My understanding is 'páirc' can mean field, but exclusively a sports field. Otherwise, 'Gort' would refer to a field..(i.e. paddock, meadow, etc., the Bull McCabe's field)
It's not exclusively a sports field. From Ó Dónaill's 1977 Foclóir:
páirc, f. (gs. ~e, pl. ~eanna). 1. Field, park. ~ féir, pasture field. ~ imeartha, playing-field. ~ aonaigh, fair-ground. ~ phoiblí, public park. Ar pháirc an áir, on the battle-field. 2. ~ éisc, large shoal of fish. 3. (In cloth) ~eanna, check pattern.
'Gort' is often used in vernacular to refer to a field in general but strictly speaking it means an arable, cultivated field, rather than one used for pasture. Hence, placenames like 'Gort a'Choirce' (Gortahork), meaning 'Field of Oats'.