I know I can use keyframes and use parent-relations in meshes in individual objects. But is it possible to keyframe or use parent relations to separate meshes in the same object (maybe when I use vertex groups)?
Use vertex groups and armatures.
If an armature can move an attached hand of a human body mesh then it can also move disconnected sub meshes of a Blender Mesh Object.
Parent the mesh to an armature. Use weight painting if necessary. Armature need not to be connected. Think of the armature as a collection of controllers and not a mechanism to control an organic mesh model.