Curbishley was Parker's coach at Charlton before the midfielder secured a big-money move to Chelsea.
But in a subsequent spell at Newcastle, he became known more for his defensive qualities than his prowess in the final third.
Now Curbishley has called on Parker to use his fine solo effort against Middlesbrough - who went ahead through David Wheater after 40 minutes before Dean Ashton thundering drive levelled the scores four minutes later - as the springboard for a productive campaign.
"I've asked Parker to be, perhaps, a bit more offensive than he was at Newcastle," the Hammers boss revealed.
"When I had him at Charlton, he could do a bit of both. At Newcastle, I feel he was a bit more responsible.
"So I've asked him he can get a bit more into the box and a pinch us a goal.
"Let's hope this is the start of something."
Curbishley's relationship with Parker was believed to have soured somewhat when he opted to leave The Valley for Stamford Bridge - a move that failed to work out for Parker - but there is now nothing but respect for the hard work the player has put in at Upton Park in the face of adversity.
"It's been such a frustrating time for Scott, getting injured in the second week of pre-season then coming back, not being quite right and going out again," said Curbishley.
"He's worked ever so hard and he fully deserves what happened (against Boro)."