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Parkinson has an excellent relationship with the ownership group, several of whom would prefer not to make a change.
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Yet results are forcing the club to consider a new manager, with new sporting director Steve Nickson involved.
TalkSPORT understands Steven Gerrard, who hasn't managed since leaving Al-Ettifaq in January 2025, is open to the role should a vacancy arise.
Scott Parker is another name expected to be considered.
Parkinson will take charge of Saturday’s game against Preston with Wrexham currently 17th in the table.
Parkinson asked for an immediate response from his players after a 2-1 defeat at home to West Bromwich Albion in midweek.
Kieffer Moore opened the scoring before Isaac Price and Aune Heggebo sealed the win for the visitors.
And after the game, he said: "When you've lost a game at home it's time to batten down the hatches and come out strong as a group, and that's what we're going to do at the weekend.
"I'm sure we'll take criticism for the defeat. That's inevitable and we totally accept that.
"But there's a lot of resilience at that training ground and we're going to show it in the build-up to Preston.
"When you lose in that manner, it's obviously harder, isn't it? Because the lads could feel it, the game was there for us.
"No-one's going to feel sorry for us in football. We've got to find a solution, look at the game and see which areas can we do better."
It has been Wrexham's inability to win games this year that has cost them.
The Welsh side have played ten, won only two, drawn five and lost three in the league.
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The Liverpool legend's first managerial role was Rangers.
He joined in the summer of 2018 and was there until November 2021 as Aston Villa came calling.
In his time at Ibrox, the Gers won the SPFL title and went unbeaten for the whole season.
That saw some have the chance to qualify for the Champions League, which they failed to do.
Instead, they went into the Europa League, where they made the final but not under Gerrard.
That was because Villa saw him as the man to lead a brave new team in the Premier League.
He spent big money on Diego Carlos, Lucas Digne and Philippe Coutinho only for things not to go so well.
His first year saw Villa finish in 14th place after picking up 35 points from 27 games before he was sacked midway through the following year.
Gerrard made it to 11 league games with just two wins and Villa in danger of going down, so he was replaced by Unai Emery.
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From there, he made a left-field move to Saudi Arabia to take charge of Al-Ettifaq.
More big-name players arrived in Jordan Henderson and Georginio Wijnaldum.
What followed was a sixth-place finish in his one full season before leaving in January 2025.
At the time of his departure, Al-Ettifaq were in 12th place, and he hasn't managed since.
Instead, Gerrard has become a regular on TNT Sports as a pundit.