Making assumptions about what happened nearly 250 years ago is a bit risky. I have assumed that the William Parker who signed Elizabeth’s marriage bond was her father. He could equally have been her brother, her cousin or no relation at all!
His job was to stand surety alongside the groom, John Johnson. They both asserted that there was no lawful impediment to the marriage. If there did prove to be a problem, they would have to pay, jointly, the sum of £100. This would be about £13,000 in 2019 value according to the excellent https://www.measuringworth.com/ so this was a serious matter.
One reason for having a licence was to prevent banns being read in the parish church, a public process that takes about a month. There could be many explanations for this, from advanced pregnancy to the wish to avoid publicity. I think the reason John and Elizabeth took this route is because the Parkers were Quakers. The Johnsons were not.
The wedding would take place in a Church of England church – the famous “crooked spire” in Chesterfield, properly called the parish church of St Mary and All Saints. Quaker marriages were acceptable and we have several examples of them in our family history, but a mixed marriage might pose some difficulties. Perhaps the Parkers did not want to broadcast the news that their daughter was marrying “out” as we would say. In this case outside the Society of Friends.
The next generation of Johnsons went even further away from the Quaker persuasion. William and Mary Parker’s grandson, another William, actually married into the clergy of the Church of England. His wife Elizabeth was the daughter of the local curate, the Rev David Holt. Perhaps it’s as well that his grandparents didn’t live to see the wedding!