Learn Hardcore Russian: Мой друг идёт в парк со мной.

идёт: one-direction, on foot, in progress now (or arranged near future).
ходит: multi-direction/habitual (goes regularly; back-and-forth).
пойдёт: perfective future “will go (set off).”
едет: by vehicle (is going/driving/riding).

Because with motion toward a destination you use в + accusative: в парк (into the park). For location (being inside), you use в + prepositional: в парке (in the park).

What case is парк here, and why doesn’t it change form?

It’s accusative singular of an inanimate masculine noun ending in a consonant. Inanimate masculine nouns of this type have the same form in nominative and accusative: парк → парк.

Why со мной and not с мной?

The preposition с becomes со before certain consonant clusters to make pronunciation easier. Since мной starts with the cluster мн-, the standard form is со мной. You may also see the slightly more formal/poetic со мною.

What case is мной, and what are the forms of я?

мной is the instrumental singular of я, used after с/со to mean “with.” Quick forms of я: nominative я, genitive меня, dative мне, accusative меня, instrumental мной/мною, prepositional мне.