It would be highly unusual to find the names of any parents or witnesses in marriage entries at this point in time.
There is no point contacting Cheshire archives.
The images are on findmypast.
Both of these marriages in the actual register just give the bride and grooms' names.
They are both by publication of banns - ie , not by licence, which possibly may have given some more information about them, although still very unlikely to give the name of the father of the groom.
Catherine Smith was buried 27th Sept 1747 at Bebbington, "wife of John Smith of Lower Bebington".
There are several burials for 'John Smith' which could be him, none of which mention 'Lower Bebington'.
BUT, a Mary Smith was buried 16th Jan 1776 at Bebingotn, 'wife of John Smith of Storton'.
Storton is just a couple of miles west of Bebington.
John Smith was then buried at Bebington 14th Oct 1780 'of Storton'.
No ages given in any of these burials.
Interestingly, on the same day as Mary Smith was buried, 16th Jan 1776, a John Smith was married at Bebington.
He married Ellen Langley. He was of Babington, and she of Great Neston. It was by banns - the dates given as 31st Dec 1775, and the 7th and 14th of Jan 1776, so these were before the death of Mary.
Possibly this was John's son John born in 1753.
Even by this date, there are no parents' names given in the marriage register.
A bit odd he should get married on the same day as they buried his mother though!
Perhaps as they had had the banns read before she died, they thought they should go through with it anyway.
I see a Mary Smith was baptised, (just 7 months after the marriage), on 18th Aug 1776 at Bebington. Father John Smith of Storton. No mother's name given. Luckily the entry in the actual register says he is 'John Smith junior'.
24th Nov 1782, baptism of Lydia Smith, daughter of John Smith of Storton.
So the burial of the John Smith in 1780, must have been the elder John Smith.