Learn Russian: Мы отдыхаем дома или в парке.

In Russian, verbs change their endings based on who is doing the action. Because the subject here is мы (we), the verb отдыхать (to rest) takes the standard "we" ending -ем.

Does отдыхаем usually mean "rest" or "relax"?

It can mean both! While translated here as "rest," отдыхать is the most common everyday verb for "to relax," "to hang out," or "to take a break." So, "We relax at home" is a perfectly natural translation.

Why is "at home" just the word дома without a preposition like в (in)?

Дома is a special fixed word (an adverb) that specifically means "at home." If you used the preposition в and said в доме, it would mean you are emphasizing being physically "inside the building" rather than just chilling at home.

Why does парке have an -е at the end?

This is the Prepositional case, which Russian uses to show where something is happening. The basic dictionary word is парк. Because we are talking about a location with the word в (in), we add an -е to the end of the masculine noun, making it в парке.