Both men scored wonderful goals, Ashton drilling home powerfully with a first-time shot just before half-time and Parker scoring in the 90th-minute with a stirring solo effort, as the Hammers mugged their hosts in what had been an otherwise even encounter.
Middlesbrough lacked the magic in the final third despite taking the lead when defender David Wheater prodded home following a clever set-up by Jeremie Aliadiere.
Hammers boss Alan Curbishley said: "There wasn't much in the game and what lifted it, I think were our two goals.
"They were two fantastic goals and they came from two players who have been chasing their fitness a bit.
"Some of the lads have had to get fit in the first team. Scotty was one of them and he had a tough time with three games in a week but he had to get through it.
"Dean was another one, we've had to throw him and say 'right, get fit in the first team'. It's very difficult when that happens."
It was a timely effort from Ashton, with many pundits tipping him to be a central figure in Fabio Capello's England set-up, but Curbishley was more concerned with decrying those who have questioned the striker's recent efforts.
"He's taken some criticism in the last few weeks and I think that is because of the expectation. He came back, he got a couple of goals (and then tailed off) but getting fitting while playing in the Premier League is so hard.
"Dean looked sharper today and I think that helped. The thing about Dean is that when he hits the ball he hits it so hard.
"When a ball drops you want it to drop to him."
Boro boss Gareth Southgate was deflated after seeing his side's three-game winning streak - their best run of form since January - ended in the dying seconds.
One of those victories came in a flowing match against title hopefuls Arsenal, while the players showed their appetite for a scrap by following that up with a 1-0 win over struggling Derby.
Southgate believes neither aspect was present against the Hammers, though.
"We never hit the heights of recent weeks today," he said.
"The quality was not as it was against Arsenal and the grit and determination was not as it was against Derby.
"We should have seen the game out and got a point. I don't think it was a lack of effort though.
"It's what separates teams at our level of the league - teams find it hard to produce week after week and that is why we are where we are.
"The big boys of this league can win when they don't play well."
That is something Southgate hopes his young side can do in the future but he admits there is currently still some learning for them to do at the end of matches.
"We're a bit naive perhaps. Yes, you want to win games but you still cannot commit everyone forward.
"In certain areas we have young players and they must learn how to win matches, or to see them out. It's a lesson they have to learn."