I think the prior answers are either misleading or wrong, and people should not miss out on the opportunity to get great DNA information because they were discouraged by those answers.
If you have the DNA for Charlie (the parent), and any of Charlie's children (Dick, Elizabeth, Frank, and George), you can at least partially phase a child (and thus pseudo-phase the parent too). An example of a tool that does this is found here:
And here is a link to a tool you should use when you have DNA for more than one child:
Ideally, you would use data from both parents to phase the DNA of the child. However, both of these tools can run with data from only one parent. The caveat is that you will probably end up with "no calls" at some of the SNP locations.
If you want to do the phasing manually, the Journal of Genetic Genealogy has an article titled "Phasing the Chromosomes of a Family Group When One Parent is Missing," available here:
By the way, you can phase Charlie's DNA using only the children. Of course, any DNA that Charlie did not pass to his children would not be reconstructed. Here is a link that explains how to phase a parent's DNA with only the DNA from the children:
These links explain the concepts much better than I can in this short answer. But here are the basic principles: By phasing the child, you will know which DNA the child inherited from Charlie. You will also know which DNA Charlie did not pass on to the child. At this point, you have already pseudo-phased Charlies DNA. Now you just need to know the recombination points for the DNA that Charlie passed on to his children. You can find the recombination points by comparing the children's DNA with each other (see the visual phasing article mentioned above).
Testing the children does not help identify new DNA connections, but it does help to determine the relationship of the existing DNA connections. First, testing can be used to phase Charlie's data, which then tells you which of Charlie's parents share the DNA with the DNA connection. Second, the phased DNA gets rid of many false matches (the IBS matches) and the remaining matches are more likely to be valid (IBD).
Hopefully you can get Charlie, Dick, Elizabeth, Frank, and George tested before it is too late. I would have them all tested, because you can probably reconstruct and phase most of the DNA of their mother(s) as well.
The original question also asked "What if some of Charlie's children (Frank and George) are with a different spouse of Charlie than the others (Dick and Elizabeth), does that help more?" The answer is yes, especially if that means you are adding a second or third sibling. There are also benefits to adding a fourth sibling, but they are not nearly as great as adding the second sibling (allowing you to identify recombination points) or adding a third sibling (allowing you to identify which sibling is responsible for each recombination point). The fourth sibling may help resolve some no-calls in the phasing of the other siblings or provide additional DNA to reconstruct a missing parent.
Edit: I was asked to provide an explanation of an actual example of phasing a parent using DNA from the parent's children.
Here is the procedure I used to completely phase my parent’s DNA. This procedure also extracted one of the two sets of 23 chromosomes (albeit pseudo-phased, not fully phased) for each of my grandparents. That means I identified 50% of the DNA (approximately 3600 centimorgans) for each grandparent.
Step 1: Mail spit samples from me, my brother, my father, and my mother to Ancestry.
Step 2: Download Ancestry DNA raw data for each of the spit submissions.
It looks like Ancestry recently changed the format of how it provides raw DNA data. I obtained raw DNA data last week. As of November 29, 2017, FamilyTreeDNA would not accept my raw data in the format provided by Ancestry. Hopefully FamilyTreeDNA figures this out and provides a solution soon. My parents obtained their DNA from Ancestry a few months ago, and FamilyTreeDNA accepted their data without any problem. I made my data file look like my parents’ data file (explained in Step 3 below), and it uploaded fine. My parents’ files are 668,962 lines long, while my file was only 650,430 lines long. It looks like Ancestry is now providing results for a slightly different set of SNPs.
Step 3: Create a new raw data file for me and my evil twin (my parents' DNA that I did not inherit). I wrote a Perl script that read in my father’s data and created an index for each RSID. The script then read in my mother’s data and added her data for each RSID. The script then read in my data. If an RSID in my data was also an RSID in my parent’s data, the script printed the alleles I inherited from parent’s to my file, and the alleles I did not inherit to my evil twin’s file. If an RSID was in my data but not my parents’ data, it was ignored. If an RSID was in my parents’ data but not my data, the script printed a “no call” (“0” instead of G, A, T, or C) for each allele in my file and my evil twin’s file.
Step 4: Upload the new raw data file for me, my evil twin, and my parents to FTDNA.
Step 5: Pay the $19 for each kit to get access to the chromosome browser. For each kit, go to the chromosome browser page and click on the link to “Download All Matches to Excel (CSV Format).”
Step 6: Copy the downloaded data from all four .csv files into a single spreadsheet. Sort the data by 1) chromosome; 2) start location; 3) centimorgans
Step 7: Use David Pike’s tool to identify the recombination points in the DNA for me and my brother.
If you have three siblings tested, it would greatly simplify the identification of which sibling was responsible for the recombination point. If one sibling’s pattern changes while the other two siblings’ patterns are the same, the one that changed must be responsible for the recombination point. With only two siblings, you have to rely on DNA matches (explained further below) from the chromosome browser data that you downloaded. I had to do it the hard way with two siblings.
Please note that Ancestry provides its data in Build 37.1 format. Accordingly, the output data from David Pike’s tool will also be in Build 37.1 format. But the FamilyTreeDNA chromosome browser data is in Build 36 format. It is important to convert the recombination point from Build 37.1 to Build 36.
Here is a link to a tool that does the conversion:
David Pike’s tool shows that either my brother or I had a recombination point on Chromosome 1 of my father’s DNA at rs6426802,rs28497538 (locations 19202507-19205672).
I entered Chr1:19202507-19205672 into the “Data” box of the conversion tool (see link above), and the conversion tool showed that the recombination was at locations 19075094-19078259 in Build 36.
I filtered the matches in the spreadsheet to show matches on chromosome 1 with a start location less than 20000000 and an end location greater than 18000000. My dad had two fairly large matches that spanned the recombination point. The first match was from 8109798-38365641 (45.55 centimorgans). That same individual matched my evil twin from 9316595-19061043 (19.52 centimorgans) and me from 19063913-38365641 (24.1 centimorgans). The recombination point is mine, and not my brother’s. It is interesting that the match start and end points don’t align with the recombination locations at 19075094-19078259. I haven’t figured out why there is a mismatch in the locations. Looking at the raw data, I think David Pike’s tool got the recombination point correct.
David Pike’s tool shows that either my brother or I had a recombination point on Chromosome 1 of my father’s DNA at 55495744-55518467. This corresponds to 54853453-54876176 in Build 36. I filtered the matches in the spreadsheet to show matches on chromosome 1 with a start location less than 60000000 and an end location greater than 50000000. My dad has a match from 38368451-62971860 (26 centimorgans). I match that individual from 38631015-62971860 (25.45 centimorgans). This means that my brother has the recombination point at that location.
Sometimes there will not be a match that spans the recombination point. You can still solve who is responsible for the recombination point by looking at matches above and below the recombination point. If I have a match on my grandma’s side above the recombination and my evil twin has a match on my grandma’s side below the recombination, the recombination point is mine. If I have matches on my grandma’s side both above and below the recombination point, then the recombination point is my brother’s.
I used the “Matches in Common” feature on FTDNA to identify the correct grandparent with the matching DNA segments.
Sometimes the recombination point cannot be refined to a single SNP. When reconstructing my parents' (and grandparents') DNA, I had to insert “no calls” for the range of possible recombination points. These ranges are usually small, and they do not impact matches if “no calls” are inserted in the correct locations.