no parish register gives surname of child (at least no 18th or early 19th C one does) - the almost standard form is A son/dau of B C and D E (the wife's surname is often not given)

The fathers name is given in the register when he accepts affiliation (or doesn't object to it) - he is then liable to half the 'running costs' of bringing up the child (see a couple of presentments under history on Manxnotebook) - if the father's name was not known at baptism then it can often be found in the presentments - I've put some 15 years of these between 1800 and 1851 on my CD - example is www.manxnotebook.com/cd_only/prsnts/1800.htm