If the family followed the English naming pattern (Not all did), I would expect Richards father to be William and Nancy’s Henry, however has they had more than one John, that might prove to be the exception to the rule. I realise you may have most of below but it might help others?
Richard Smith Marriage 19 September 1775 Bidstone, CheshireSpouse: Nancy Linacer
Baptisms Bidstone.
Mother NANCY: William 1783, Henry 1885, Nancy 1789, John 1788 & 1791,
Mother ANN: Mary 1776, Peggy 1777, Betty 1779, Kitty 1781.
This marriage 1792, so I have ignored later baptisms at Bidstone
Richard Smith Marriage Notice 10 September 1792 Cheshire Spouse Ann Noble
Richd. Smith Burial 18 October 1820 Bidstone, Cheshire Birth 1755 (No age)
Nancy Linacre buried 16 October 1842 age 88
1841 -
https://www.familysearch.org/ark:/61903/1:1:M7M1-C611841 – Nancy & John?
https://www.familysearch.org/ark:/61903/1:1:M7M1-4C7Colin