Despite an early brace of tries from Shane Williams and an effort from Lee Byrne, which was set up by young fly half James Hook, a try by Parks on the half time mark kept Glasgow in contention.
Glasgow's mounting pressure on the resumption of the game told when they secured their second try of the game. Graeme Morrison sneaked over in the corner and Parks's conversion from the touchline equalled the scores at 26 apiece.
A 67th penalty from the boot of Parks nudged Glasgow into a narrow lead and as they closed down the game in the final minutes the Scottish side secured their fourth league victory in a row.
Glasgow: Tries: Parks, Morrison. Cons: Parks (2). Pens: Parks (5)
Ospreys: Tries: S Williams (2), Byrne. Con: Hook. Pens: Hook (2), Connor
Glasgow: Lamont, O'Hare, Henderson, Barrow, Evans, Parks, Beveridge, Va'a, Thomson, Low, Newman, Kellock, Swindall, Barclay, Beattie
Replacements: Morrison for Barrow (54), Tkachuk for Va'a (54), Lawson for Thomson (54), Wilson for Swindall (72)
Ospreys: Byrne, Walker, Parker, A. Bishop, S. Williams, Hook, Marshall, D. Jones, Bennett, A. Jones, Cockbain, A. Jones, Thomas, Pugh, Tiatia
Replacements: Vaughton for Walker (78), Connor for A. Bishop (72), Spice for Marshall (62), Griffiths for A. Jones (73), Powell for Cockbain (41), Lloyd for Tiatia (69)