The Pilgrims were holding on after Reuben Reid had fired home his 19th goal of the season in the 32nd minute.
But Aldred, who had scored his first league goal on Tuesday, headed home his second in the 81st minute to secure a point for relegation threatened Accrington.
The game was in doubt before kick-off due to persistent rain and the pitch was a quagmire.
Plymouth started well with the prolific Reid denied by the legs of Stanley goalkeeper Marcus Bettinelli before the striker opened the scoring.
Reid received the ball on the left side of the area and drilled it home low and hard into the bottom corner.
The game was end-to-end with both side's creating chances and Stanley substitute Kayode Odejayi should have done better when he headed over while unmarked.
Neal Trotman also failed to find the target when in a decent position for the visitors before Aldred equalised, powering home Lee Molyneux's right-wing cross nine minutes from time.
Both sides went for the winner in an entertaining clash but had to settle for a share of the spoils.