The rules had been set out in Hardwicke's Marriage Act 1753.
The Act required that, for a marriage to be valid, it had to be performed in a church and either after the publication of banns or the obtaining of a licence.
If the marriage was by licence, those under the age of 21 had to prove parental consent before the licence was issued.
If the marriage was by published banns, then express proof of consent wasn't actually required. If the parent wished to prevent the marriage, they could simply forbid the banns.
Note that the parent had to actively forbid the banns in order to prevent the marriage. This might be straightforward if the banns were published in the couple's home parish. However, some had their banns read in parishes where they were not resident, and where their parents were thus unlikely to see them. In those cases, the marriages were still deemed to be valid.
From the Wikipedia page on Hardwicke's Act:
While the parent of a minor could forbid the banns and so prevent a marriage from going ahead, a marriage by banns that took place without active parental dissent was valid. This gave rise to the practice whereby underage couples would resort to a parish where they were not resident to have the banns called without their parents' knowledge. Since the Act specifically prohibited the courts from inquiring into the parties' place of residence after the marriage had been celebrated, such evasive marriages were still valid.