I would look at the introduction in
and then the "Using MathObjects" that you are looking at (that is also linked to the page above), then the "MathObject-based answer checkers" linked there. Then look at the context information, which is much sketchier.
As for Parser.pl and MathObjects.pl, use MathObjects.pl in your problems. The Parser.pl file is the same as MathObjects.pl, but it was renamed when the MathObjects were given their official name (the Parser was their older name).
Loading MathObject.pl will get you access to the standard MathObjects and contexts. There are additional contexts in /pg/macros in files starting with "context" and additional MathObjects in files starting with "parser". You need to load them individually when you need to use them.