A search of the British Newspaper Archive produces this fragment from the Staffordshire Advertiser of 14 October 1916, which suggests that he was in the Royal Garrison Artillery at the time of his marriage, OCR issues aside. Obtaining a copy of his 1916 marriage registration would be quite useful at this point I think.
The wedding of pte. Parrott, Royal Garrison Artillery, fourth son of the lat Mr. Frank Parrott, of Seig) ford Grange and ford Hall, Seighford, snd Miss Deis, Mary Tie ...
There are medal index cards for at least two Frederick Parrott's serving with the Royal Garrison Artillery, one with service numbers 850 and 292554, the other with service number 74913.