Marriage registration began in 1837 and there doesn't appear to be a marriage of ATKINSON/PEARCY between 1837 and 1841 in the GRO marriage indexes. Earlier marriages will only be in parish registers,and to find those you are going to have some idea of where the birth of William was - I would concentrate on that.
There are no births of a William Atkinson showing PEARCY as the mother's maiden name in the GRO birth index for that period. But there are a few entries worth further investigation ...
There are two William ATKINSON's with mother's called PEARSON, one born in Scarborough in 1837, and one in Durham in 1839.
There is a Charles ATKINSON born in Sculcoates in 1838 with mother's name of PIERCY which is also a family you should look at.