The following are some of the common definitions for part
- a piece or portion of a whole
- an integral constituent of something ⇒ dancing is part of what we teach
- an amount less than the whole; bit ⇒ they only recovered part of the money
Both part and a part can be used to refer to a discrete countable thing
We teach several skills. Dancing is part of what we teach.
We teach several skills. Dancing is a part of what we teach.
Dancing is a discrete thing but can take either construction.
And both part and a part can be used to refer to a more general portion of a thing that is not countable
Part of the food [or money, or care] will be devoted to the poor.
A part of the food [or money, or care] will be devoted to the poor.
The portion to be devoted is not defined in these sentences as comprising a specific discrete section or item, but either construction is clear and appropriate. The addition of the article a may give a slight connotation that the portion may already be determined, but not necessarily.
The term parts, on the other hand, almost always refers to discrete items.
Speaker A: Parts of our efforts have been highly successful.
Speaker B: Which parts?