Eyebrows were raised in the lead-up to kick-off at Anfield after it was announced that Brazilian duo Philippe Coutinho and Roberto Firmino would both be starting on the bench for the hosts, with Klopp making six changes to the team that thrashed Spartak Moscow in the Champions League.
Despite enjoying nearly 80% of possession, the Reds could only come away with a point after Wayne Rooney's late penalty cancelled out Mohamed Salah's opener.
Thompson said Klopp's team selection suggested that he did not grasp the importance of the Merseyside derby to Liverpool fans.
"We, as Liverpool fans, have moaned about the rotation for a few weeks, and it has come off previously, but this is a derby game. You play your best team in the derby," he told Sky Sports.
"That selection worried me that he didn't understand how big a game it was. He's gone with a team to beat a mid-table team rather than win a derby game.
"If it was Manchester United, he would have played a full-strength side, so it was maybe a bit of complacency as well as the rotation.
"It's always a risk rotating and, for this game, I just can't get my head around it as a Liverpool fan."
Klopp also lashed out at a Sky Sports reporter in the post-game interview after an argument erupted over whether Dejan Lovren's push on Dominic Calvert-Lewin was worthy of a penalty offence.
Thompson defended Klopp's reaction, noting that emotions were running high after a tense derby game.
"I've been there, it's frustration and anger. His team had so much possession but Jordan Pickford wasn't really worked," he said.
"I can understand and he did apologise but I don't think he's had a good look at it because he wouldn't have treated the reporters that way. Yes, it was very harsh and the referee couldn't wait to point to the spot, but it is one of them we see given."