Parks' points-harvesting cost the Ospreys dear
Glasgow: (16) 29
Tries: Parks, Morrison
Cons: Parks (2)
Pens: Parks (5)
Ospreys: (26) 26
Dan Parks scored 24 points as Glasgow clawed back a 17-point deficit to put a huge dent in the Ospreys' title hopes.
Tries: S Williams (2), Byrne
Pens: Hook (2), Connor
A first-half brace from Shane Williams and a Lee Byrne score set up by James Hook put the visitors in command.
But inconsistent kicking from Hook cost the Ospreys, and a Parks try on the stroke of half time kept the tie alive.
A sustained Glasgow attack was rewarded with a Graeme Morrison try, Parks' conversion drawing things level, then a 67th-minute, 50m Parks penalty won it.
The victory was the fourth league triumph in a row for the in-form Scots and the Ospreys' first defeat in the competition in seven games.
The visitors had a dream start at a sandy and parched Hughenden, Williams presented with an overlap from the first move of the game and taking the outside of Thom Evans for a lightning try.
Our goal this year was to get the respect of the Welsh and Irish sides
Glasgow coach Sean Lineen
James Hook converted from the touchline, and added a penalty against two successful kicks from opposing outside-half Parks.
But it was the Ospreys number 10 who was lighting up the field, picking the narrowest of gaps in midfield with a devastating break before sending Byrne under the posts.
Hook somehow missed the simple conversion, though, an energetic charge from Rory Lamont shaking the half-back's usual composure.
Glasgow fought back, prompting a string of Ospreys indiscretions that resulted in a sin-binning for Adam Jones, Parks landing the resultant penalty from under the posts.
Hook knocked over a difficult kick to extend the lead to 18-9, then, as the home side increased the pressure on the depleted Ospreys, Parks threw an interception pass that was seized upon by Williams.
The Wales wing pounced, outpacing Hefin O'Hare from 80m out for a try that Hook again failed to convert.
The outside-half went off temporarily with a head injury, and with his first touch replacement Shaun Connor added a penalty.
A quickly taken penalty saw Glasgow mount a sustained attack that ended with Graeme Beveridge setting up Parks from short range, the latter's conversion reducing the gap to 26-16 at the break.
Wales summer tourist Brent Cockbain went off at half-time with a leg injury, and Parks went on to bring the home side back to within a score with his fourth penalty.
We looked a very tired team and buckled under too much pressure
The home side encamped on the Ospreys line as the visitors went into their shells, and the reward finally came as Morrison sneaked over in the corner.
Parks converted from the touchline, then added the long-range winning kick after Alun Wyn Jones offended at a ruck.
Glasgow closed the game down effectively in the final minutes, Hook missing touch with a late penalty to try to set up a winning position.
The Ospreys will now have to rely on results involving the Blues and Leinster going their way, and can only concentrate on winning next week's games at the Dragons and Border Reivers.
Glasgow coach Sean Lineen:
"It was a hard-fought, gutsy game from both sides.
"This team has come a long way this season, there are a lot of young guys in it and I'm delighted with this result against a side as strong as the Ospreys.
"Their belief is huge, they have a passion to play for Glasgow, they are excited by what they're doing and they never give up.
"Our goal this year was to get the respect of the Welsh and Irish sides and we're slowly getting there.
"There are exciting signings to come in the summer and things are happening here."
Ospreys coach Lyn Jones:
"We looked in control at the break and are very disappointed not to come away with a win, but there are no excuses.
"Glasgow played well and made few mistakes, while we looked a very tired team and buckled under too much pressure.
"They just kept creeping up on us and at the end closed the match off, they've learnt how to win games.
"League destiny is out of our hands now, with Leinster back as favourites while we have two very tough games next week.
"All we can do is turn up at the Dragons and concentrate on our performance. They will be as tired as us."
Glasgow: Lamont, O'Hare, Henderson, Barrow, Evans, Parks, Beveridge, Va'a, Thomson, Low, Newman, Kellock, Swindall, Barclay, Beattie.
Replacements: Morrison for Barrow (54), Tckachuk for Va'a (54), Lawson for Thomson (54), Wilson for Swindall (72). Not Used: Eddie, MacFadyen, Gregor.
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). Not Used: Hibbard.
Referee: George Clancy (IRFU).