Cameron Carter-Vickers scored the only goal of the game in what was a tight contest at the Macron Stadium, leaving Wanderers rooted to the bottom of the Championship table with just two points to show from their opening seven fixtures.
Speaking afterwards, Parkinson refused to criticise his side and instead singled out Attwell, claiming that he had failed to control the game in the second half.
"I don't think we could have given much more," he told The Bolton News. "We gave everything to win the game and then get back into the game when we conceded the goal from the scrambled corner.
"It would be nice if we got a ref who gave us a few decisions but everyone has seen his performance, which for me was a very disappointing one on the night.
"I thought his performance was very poor. When he got into the Premier League and got knocked back down we had him a couple of times and I thought he had a good demeanour, knew the game, but I thought he went backwards there.
"I think he got a lot of calls wrong for both teams and I spoke to Chris (Wilder, Sheffield manager) about it, and he agreed it was a frustrating performance."
Next up for Bolton is a trip to Ipswich Town on Saturday afternoon.