Learn Spanish (Latin America): Vamos por el parque.

Even though you are learning to use en for location, en is mostly used when you are stationary in a place (like Estamos en el parque for "We are in the park"). When you are talking about motion through or around a space, you use por.

Why isn't it Vamos al parque?

It changes the meaning! Vamos al parque means "We are going to the park" where the park is your final destination. Vamos por el parque means you are moving through the park on your way somewhere else, or just wandering around inside it.

Could por mean anything else besides "through" here?

Yes, when talking about movement, por is very flexible. Depending on the situation, Vamos por el parque could also mean "We go around the park" or "We go by the park."

I don't see the word for "we" (nosotros) here. Is that normal?

Yes, it is completely normal. The verb vamos is the "we" form of the verb ir (to go). Because the verb itself tells us exactly who is going, you can safely drop the pronoun nosotros.