In an incident-packed game, Inter looked to be heading for a much-needed win thanks to a Mauro Icardi brace which turned the scoreline around following Luca Toni's early opener.
Mazzarri's side were reduced to 10 men soon after half-time when Gary Medel was given a second booking for handball in the area but Samir Handanovic thwarted Toni to register his third spot-kick save of the season.
But Verona substitute Nicolas Lopez ensured there would be no reprieve with an equaliser two minutes from time that leaves Inter ninth in the table and winless in three games.
Media reports have linked former defender and current Sampdoria coach Sinisa Mihajlovic with the job but Mazzarri saw enough to give him hope for the future.
He told Sky Sport Italia: "We had a great first half, though we were punished at the first mistake on the counter. It could've been a heavy blow, but we reacted, turned the game around and were unlucky not to go 3-1 up.
"I thought we dominated the game for about 70-80 minutes, though of course it was normal to struggle when going down to 10 men. The fitness levels also fell away as we are in an injury emergency situation.
"We still managed to go round them and put dozens of crosses into the box. You need patience in those situations and I thought Inter played a great first half.
"On the equaliser we allowed Verona to go round us. We are accustomed to attacking and not sitting back, while we were also exhausted and let Verona move the ball around in the box.
"It's normal that when a game runs away from you like this and the incidents go against you like that red card, it all becomes more difficult. We have so many injured players and this break will hopefully help us achieve acceptable fitness levels again.
"We are back in Europe this season, which means playing every three days, and have undertaken a project of helping young players to grow.
"Of course when fans are accustomed to being at the top, it can pile pressure on. It's one thing to grow young players when you aim for normal targets and another when you go for bigger things.
"It's not that bad. The team had a couple of bad performances, but they are united and play good football, so I have reason to be optimistic."
