In the 1841 census they live a few doors away from a Wilson family- maybe one of them is Jessie's dad?
Looking at the 1881 marriage cert of Eliza Perren,it lists her as a spinster,so it sounds like she had Jessie as a single mum-the fathers name may well have been Wilson though.
It wasn't compulsory to register births before 1875 so not everyone did.
Carol