Learn Russian: Мы спали в тесной палатке, в то время как наши родители отдыхали на берегу. - We slept in a cramped tent, whereas our parents were resting on the shore.

It is acting as a conjunction of juxtaposition, meaning 'whereas.' Even though it literally translates to 'in the time how,' it is used here to contrast the less comfortable situation of the speakers (in a cramped tent) with the relaxing situation of their parents (on the shore).

Can I split the phrase в то время как with a comma, writing it as в то время, как?

Not in this sentence. When в то время как is used as a single conjunction of contrast ('whereas'), the comma is placed entirely before it. You would only put a comma before как if you were pointing to a literal specific time ('at the exact time when...'), which isn't the case for juxtaposition.

Why does the word for shore (берег) end in -у in на берегу instead of the usual prepositional -е?

The noun берег is one of a special group of short masculine nouns that take the locative ending -у (which is always stressed) when answering the question 'where?' with the prepositions в or на. Other common examples you might remember are в лесу (in the forest) and на полу (on the floor).

Why use the imperfective aspect for both спали and отдыхали?

Both verbs describe continuous, ongoing actions happening simultaneously. Conjunctions like в то время как naturally pair with imperfective verbs when painting a background picture of two parallel states occurring at the exact same time.