Learn Russian: Папа и мама дома.

The conjunction и simply adds things together, like "Dad plus Mom." They are both doing the exact same thing (being at home). You would use а if you were contrasting them, for example: Папа дома, а мама на работе (Dad is at home, while Mom is at work).

Why is there no word for "are" in this sentence?

In Russian, the present tense of the verb "to be" (am, is, are) is almost always skipped. You just put the subjects (Папа и мама) right next to their location (дома).

Why is it дома instead of just дом?

The word дом is the noun meaning "house" or "building." When you want to say someone is actually "at home" or "in the house," you must use the special location word дома.

Does it matter which person comes first before the и?

Not at all! Just like in English, the order of the items connected by и doesn't change the grammar or meaning here. You can easily say Мама и папа дома.