My initial thought was to suggest that you order one (or both) certs from the Bham register office. However, looking at the dates, I've had a look at the 1901 census and they are at 26 Vincent Crescent, Balsall Heath (Ecclesiastical parish of St Paul's, Civil Parish of Balsall Heath). This comes under Kings Nortong Reg district, so they moved after Thomas was born - so I think that you will have to order certs if you want to know where they were living when they were born - unless you trawl through all the possible parish registers.
Full details of census:
Cornish James A , Head, Married, 36, 1865, Ornament Mounter, b Birmingham
Cornish, Catherine, wife, Married, 36, 1865, Seamstress, b Ripley, Surrey
Cornish, James A Son, single, 6, 1895, b Birmingham
Cornish, Thomas H, son, single, 3, 1898, b Birmingham
Hope this helps a bit