Are mesophyll cells the same as parenchyma cells?
Despite countless sites and other resources saying otherwise, Mesophyll is not a tissue technically speaking. It's just a place, a specific area. Or you can even think of it as the set of all tissues in a given area.

And that given area is easy to define: the "filling" of the leaf, between the upper and lower epidermis. According to Mauseth (2012), that with Raven is my favourite book on Botany:


  Mesophyll: All tissues of a leaf except the epidermis.


Actually, this is easy to see if you understand the Greek origin of the term: μέσος, meaning "in the middle", and φύλλον, meaning "leaf". Therefore, mesophyll means "in the middle of the leaf".

In a common, regular leaf, most of the mesophyll is made of parenchyma, be it palisade or spongy. As I said before several sources treat these terms (mesophyll and parenchyma) as synonyms, but they are not.

Just for completeness, this is a very simple list of the plant tissues. You'll see that mesophyll is not on it:


Meristems
Dermal tissues


Epidermis
Peridermis

Ground tissues


Parenchyma
Collenchyma
Schlerenchyma

Vascular tissues


Xylem
Phloem





Source: Mauseth, J. (2012). Botany. Sudbury: Jones &amp; Bartlett Learning.