Mistakes are part of the learning process.
This is the best, most natural choice. Even though you are using the definite article 'the', this is understood to be general, or mean "most or all learning processes".
Mistakes are a part of a learning process.
Mistakes are part of a learning process.
These two are both intelligible and correct grammar, and have the same meaning as does the sentence with 'the'. However, they can sound a little bit strange on their own, without context, and the first might make us want to identify a specific learning process. You will not go wrong using 'the' instead as a general rule. Here are examples of "natural" uses for both:
Mistakes are part of a learning process, and this activity is part of a learning process!
We'll try not to make mistakes, but mistakes are a part of a learning process.
A speaker might add time-stress to "are" in the second sentence above, which conveys almost exactly the same connotation as would adding ", after all" at the end - an emphasis that implies the listener should expect mistakes to happen.
Mistakes are parts of a learning process.
is incorrect use, though not incorrect grammar - a listener might be confused. Saying it this way makes us think of a container / contained, or comprising or forming relationship, and not of the a composing aggregate / whole aggregate relationship. Don't use it to mean what you want to say here.