Altrincham have been hit by positive Covid-19 tests this month which saw their recent games against Solihull Moors and Bromley postponed, with a number of squad members self-isolating for a spell.
They managed to play at the weekend, beaten 2-1 at AFC Fylde in the Emirates FA Cup fourth round qualifying tie and had hoped to push the game against Pools back 24-hours following the recent disruption.
Parkinson stressed Pools were well within their rights to stick to the original date but has said he was still disappointed.
The game at Victoria Park gets underway at 7.45pm on Tuesday behind-closed-doors.
Speaking to his club website ahead of the trip to the North East, Parkinson said: “It’s going to be a really tough game, every game we have got is going to be a massive challenge for us.
"We will be underdogs in the vast majority of games we go into, we punched above our weight and got up last season and this season we will have to do it again against a well-run team with exceptional players.
"It was a little bit disappointing in terms of their attitude not to move the game to a Wednesday with what we’ve been through but that is their prerogative to stick to a Tuesday and make it difficult for us, we will certainly remember those types of things moving forward.”
He added: “When everybody is facing this epidemic you expect football clubs to stick together, Fylde were top drawer in moving the fixture, realising the situation we were in and we appreciate that.
"I wish them all the best, they look favourites to win promotion. They are doing everything right to move in the right direction.
"It is absolutely Hartlepool’s prerogative not to change but it is slightly disappointing with the current climate and what everyone is facing.”
Pools boss Dave Challinor doesn’t have any fresh injury concerns ahead of the National League clash back on home soil, with Pools in good form on the back of their 6-0 FA Cup win at Ilkeston Town at the weekend.