First-half goals from Philip Billing and Dominic Solanke proved enough for Parker's side, despite Reece Burke pulling one back in the second half.
"We're undefeated and that's no mean feat in this division," said Parker who praised the contribution of recent arrival Ryan Christie who chipped in with both assists.
"I'm pleased with him, he's done well. He's got that craft that you need in those tight spaces between the lines - that nice bit of class."
Bournemouth had to survive a late fightback after Burke pulled a goal back for Luton in the second half.
"At that point it's about character and mentality and doing the basics and I felt we did that. They made us fight," said Parker.
It was Luton who threatened first with Elijah Adebayo having two excellent chances to put the Hatters in front after a quarter of an hour.
The first came as Harry Cornick burst down the right and squared the ball to Adebayo who somehow managed to miss an open goal as the ball deflected from one foot onto the other.
Moments later, the powerful forward charged down an attempted clearance from Bournemouth keeper Mark Travers who was glad to see the rebound run kindly for him.
Parker released Adebayo when manager of Fulham so the 23-year-old would have been especially frustrated not to have made his mark.
Luton boss Nathan Jones rued his side's missed chances which proved costly.
He said: We had the best chances of the game really. We put it over from two or three yards and you can't do that at this level.
"Second half I thought we were excellent, really front footed but we've had to chase too many games lately. We're in the top three in chances created but we're just not converting them at the moment."
A clever move involving Solanke and Jefferson Lerma saw Christie deftly lob the ball into the path of Billing who scored with a side-foot volley.
Parker's side were in cruise control as they doubled their lead after half an hour. Christie was once more the architect, chipping the ball up invitingly for Solanke who planted a header down into the corner.
They might have been three up by half-time had Lerma capitalised on a slip from Luton keeper Simon Sluga - the Colombian could only drag his shot wide.
There were a few signs of a fightback from Luton at the beginning of the second half. Travers saved well for Bournemouth after Henri Lansbury's whipped free-kick was diverted towards his own goal by Adam Smith.
They gained a foothold in the game when Luke Berry got down the right flank and slid the ball across for Burke to bundle it over the line via Bournemouth's Lloyd Kelly but were unable to fashion an equaliser.