New hooker Jamie Blamire scored three of Leicester’s 12 tries against Nottingham, but Parling felt his side occasionally “compounded” their errors.
“We really need to push on,” said the 41-year-old. “I’m not pleased with shipping as many points as we did.
“It’s satisfactory overall, maybe. Leicester Tigers fans want to see a team that is organised and works hard for each other.
“I’d hope we’d have a strong set piece, a decent defence and work on the back of that. There is work to do in all of those areas.”
Parling, who won two Premiership titles as a Tigers player between 2009 and 2015, was assistant coach with Australia until August, working with the Wallabies for the series against the British and Irish Lions.
“I finished that last Test on Saturday, went back to Melbourne on the Sunday and we flew as a family to Leicester on Monday,” he said.
“We landed on Tuesday and I started work on Wednesday. It’s been a whirlwind and I’ve not really had time to step back and reflect at all.
“I’m enjoying it so far. I love this club and the community. The people have been super-friendly.
“Club and family-wise, it’s been a positive start. It’s an iconic stadium and club and I’m very proud to be representing it.
“My wife would say I’m never happy with anything. I certainly think we’ve got a hungry group of players.
“We need to be more consistent in our application to the basics. We need to do it every time, not just some of the time.”
Leicester host Saracens on Saturday (15:00 BST) and visit Harlequins on 19 September (19:45) in the Premiership Rugby Cup before starting their league campaign at Bristol Bears on 28 September (15:30).
