Coral Jones, 41, said: "It's disgusting that Bridger could get money after what he has done. He's killed a little girl and ruined our lives and he wants money for a cut on his face, he shouldn't get a thing. Where's he going to spend it anyway?"
The 47-year-old, who was sentenced to life for the abduction and murder of five-year-old April, was slashed across the face by another inmate with a makeshift knife.
Now he has told lawyers to get him a payout for emotional and physical distress he claims he suffered, according to The Sun.
Bridger is not believed to have demanded a set amount - but similar cases brought by serving convicts have landed them around £15,000.
By contrast, April's parents Coral and Paul Jones can expect a maximum of £5,500 each for their daughter's murder, a total of £11,000.
They have not received a penny from the Criminal Injuries Compensation Authority since April's murder in Machynlleth, Powys, in October 2012.
Bridger is locked up at Wakefield prison, West Yorkshire. He had only just been allowed to mix with other inmates when convicted murderer Juvinal Ferreira slashed him with a Bic razor blade. He needed 40 stitches.
A Prison Service spokeswoman said: "We robustly defend all compensation claims from prisoners."