Depending on the DVD player (or TV), there are a few buttons that won't have a negative impact. I always suggest trying to find if your remote has a display button. My Panasonic DVD has one that just changes the info on the LED display (no on-screen). That would work (but my Panny doesn't have parity bits :O) Many use the Angle button, but you can also try a discrete "ON" command if available. IF your player is already on, it would accept the command, but ignore it completely. For the TV, some original remotes work for more than one TV with extra buttons for different functions. Even if the function isn't available on your TV, the command may still be accepted, and ignored, thus changing the parity status.

You need to be creative, and learning the micro-macros may take more than one or two tries, but it's cheaper than buying a new remote.