The powerful forward reportedly sat out full training this week after suffering injury against Blyth Spartans on Tuesday.
The 2-0 win was a big relief for manager Gary Mills but another test awaits today.
City face a big Yorkshire derby game with Bradford Park Avenue in an all-ticket encounter today but they may have to make the short trip without Parkin.
Mills said the player is "50-50" and will have to make a decision on the morning of the game.
Amari Morgan-Smith could get the nod if he doesn't pull through a fitness test.
Manager Mills knows the size of the game and said: "Every game is a big one but the fact it's a derby will make it even more difficult.
"It was important we got back to winning ways quickly after losing to Telford.
"We now need to make it two wins from two if we can but it's going to be an incredibly challenging match."