Parker was sacked on Tuesday, having said the Cherries' squad is "ill-equipped" for top-flight football in the aftermath of Saturday's record-equalling 9-0 drubbing at Liverpool.
The south-coast club produced a resilient response to their Anfield embarrassment and the change of manager, albeit winless Wolves wasted two excellent late chances at Vitality Stadium and also hit the crossbar before being jeered off by the travelling support.
Caretaker O'Neil said he has not "had time to breathe" during a whirlwind few days but was proud of the efforts of his dogged side and is optimistic for the future.
He said: "Did I expect my first game (in management) to be in the Premier League off the back of a tough defeat? No, I didn't.
"I had 36 hours to think about it and I used all 36 hours thinking.
"The underlying feeling is proud of what they produced and hopefully we can improve certain bits.
"It all moved pretty quick and I haven't really had time to breathe or assess where we are.
"As a group, they're the most honest, hard-working, together, relentless group that I've ever been involved with, even in my playing career.
"You can throw everything at them and they will never quit. And I think we have enough to be competitive at this level."
Bournemouth parted company with Parker just four games into the new season and three days on from his frank comments following the humiliation on Merseyside.
Cherries owner Maxim Demin cited conflict over club strategy and concerns about respect when dismissing the man who masterminded last season's promotion, rather than a knee-jerk reaction to the joint worst defeat in Premier League history.
O'Neil is unsure how long he will remain in the hot seat but will now begin preparation for Saturday's trip to fellow newly-promoted club Nottingham Forest after halting the team's three-game losing streak.
"It was a tough result at the weekend and changes in the week," continued O'Neil. "To be able to come out and show the fans, who were excellent, that we're still together was a big positive for us.
"Every point in the Premier League, especially when you're a newly-promoted team, is a hard-fought point."
Frustrated Wolves boss Bruno Lage insisted he will not put pressure on new signing Sasa Kalajdzic to solve his struggling side's problems in front of goal.
Wanderers dominated in Dorset but have now drawn blanks three times this season as their winless streak stretched to 12 top-flight games.
Austria international Kalajdzic completed a £15million move from German club Stuttgart earlier on Wednesday and could be available for the weekend clash with Southampton.
"We are very happy, not just about him," said Lage, who saw Raul Jimenez and Daniel Podence squander late chances after Matheus Nunes hit the crossbar early on.
"I will not put pressure on him just to score goals; the goals should come from everywhere. It's not just a one-man job.
"We didn't score, we are frustrated. Now is not the time to cry for the chances we didn't score. I think we deserved more.
"We know we need points, points come from goals, goals come from good performances and chances like we created today.
"I can't score goals but I can prepare the team to play with a dynamic that can create chances and score goals."