What is the difference between 'procession' and 'parade'?
While both include numbers of people walking together in somewhat of a formation, there are definite differences.

A procession is usually part of a formal occasion, such as a funeral procession, or a wedding procession.

A parade is a celebratory staged event that usually includes bands, floats, etc.

Parade is used in a literary sense, also, such as in, "watching the parade go by," which can be a literary reference to watching life pass you by without participating.
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Would you say that 'parade' is a type of 'procession' and not the other way around?
Yes, I would agree with that. A parade is a type of procession, but when one uses procession, one would not think of a parade. (The exception might be in New Orleans where they have those wonderful Jazz processions for funerals that are very akin to parades! But, then that is NOLA, and everything is an exception in NOLA!