Yes - they could. Some RC parents might have wanted the child to go to a school close by (if the RC school was a distance from home) - not sure they would have chosen a C of E school though - not in those days. By the way I can't see a church behind the school on the map in 1916. St Johns church and school were much further down the road address given as George Street.
ive just checked again and i cant see a church behind the school either jan..think i may have miss understood post 17 on this thread
https://birminghamhistory.co.uk/forum/index.php?threads/mary-street-school-balsall-heath.46391/