Charles wrote: "Are there inexpensive parts cleaners on the market that can be used to clean bicycle parts?"
I echo John's comments about older methods, like thinner (mineral spirits). But suggest (pretty strongly) against using gas - or lighter fluid or any other such exceptionally flammable stuff. If you'd ever seen the results of someone's hands catching fire after that treatment, you'd agree it's not a hollow warning.
Personal experience also says be very careful with the citrus cleaners, more for the parts than the person. Test it on less visible patches because I've seen it eat some paints, dissolve some plastics, and really mess up some decals.
Key is to think through what you need to clean: if it's an oil based grime, build up of years of grease & lube oils, mineral spirits will do it; if it's dirt, water & maybe a little dishwashing soap used with that old toothbrush does it. I'm more for the patience side of things, and "elbow grease", after having had impatience and solvents, well, dissolve things I didn't want dissovled.
Once major gunk is off, I've had great luck with the various 'soft scrub' products that clean sinks without messing up the porcelain or composite coatings. That and then a final rub down with Meguiar's (or similar) low-cut car scratch remover or polish on frame bits and it gets rid of oxidized paint & brings the finish out nicely.
Bob Hillery (how do you avoid mistakes? Experience. How to you get experience? Make mistakes ... or listen & watch when others do!) In Stratham, New Hampshire, US