"I eat between my father and mother."
In this case, being 'between' two people or objects, it would be あいだ. As a standalone (non-compound) character it would normally be あいだ, but not always. It can also be read as ま and かん. These readings would have different meanings.
・間（ま）に合う Make it in time. ・その間（かん）In that period ・その間（ま）In that (physical) space ・その間（あいだ）In between people or things.
When dealing with periods of measurable time, in compound kanji words, it would be かん (時間 - time, 期間 - period, 年間 - year).
Generally, in almost all cases the form 'A と B の 間' would be pronounced あいだ. The other readings you will likely get into eventually, but don't concern yourself too much about them right now.
De The particle de is used for the place where an action takes place. For example,
Gakkō de benkyō suru (学校で勉強する), "I study at school" Kōen de ohiru wo tabeta (公園でお昼を食べた), "I ate my lunch in the park". Toshokan de hon wo yonda (図書舘で本を読んだ), "I read a book in the library". Ni
The particle ni is used for:
(1) A place where something exists Kouen ni ahiru ga iru (公園にアヒルがいる), "There are some ducks in the park." Hon wa kaban ni haitte iru (本は鞄に入っている), "The book is in the bag." (2) To indicate a destination Watashi wa mainichi kaisha ni iku (私は毎日会社に行く), "I go to the office every day." Nōto ni ji o kaku (ノートに字を書く), "I write in my notebook."
で is used for location where an action is performed, in this case 'eating'.
に is used for simple location with non-action verbs (with motion verbs like 行く it is used to indicate direction and placed after the location headed towards).
For verbs that can be either transitive or intransitive (like 座る), there might be some overlap depending on whether the location or the action is being emphasized.