Parker, the man of the match, acknowledged the League One side had their fair share of pressure but could not overcome the two-goal deficit.
"I don't think it was as comfortable as it seemed," he told ITV Sport.
"We were under a lot of pressure at times. Once we scored we needed the second one and luckily we got it quickly. It took the sting out of them a bit and from there we went on to win it comfortably.
"We'll be looking onwards and upwards from here."
Hartlepool manager Chris Turner was delighted with his team's effort but was left to rue the penalty decision which gave the Hammers their second goal.
"There were a number of key issues in the game which didn't really go our way," said Turner.
"To get to 43 minutes at 0-0, I was delighted with the way we were playing. But they had one half-chance in the box which they finished and that's what you expect from a Premier League team.
"But then the second ball came in a minute later was handball but quite clearly outside the penalty box. It gave us a mountain to climb, especially against such opposition.
"Those players have done fantastically well. We could have been beaten by four or five but we came back out and took the game to them.
"But for a fantastic save from (Robert) Green it could have been different."
Turner added: "I'll give credit where credit is due. West Ham have come here and done a very, very professional job. But our boys have put on a great show."