Adding "on" converts this sentence from US English, which omits certain prepositions (a possible influence of German Americans on the language of the USA), to UK English, which doesn't.
Whilst I'm here: "leaves" is ambiguous in this sentence.
If the intended meaning is "begins his journey", it's safer to use the unambiguous "depart", because otherwise the subtext is:
"Your Dad leaves on Friday" to live with Gran and Granddad for a while. He needs some time by himself to think things over. – No, we're not getting divorced...
While I am less informed about the difference between US and UK English in the use of prepositions, I am - being a native speaker - pretty sure about the use of prepositions in German, and I may tell you that to omit prepositions referring to temporal nouns such as days of the week is by no means standard German. It is very much in use, though, in Berlin and the surrounding area. Having said this, let us remember: this is a Polish course.
Whatever is after "w", it must be either in accusative or in locative.
Days of the week, times of day, names of holidays, are in accusative:
W środę, w piątek, w sobotę, w południe, w wakacje, w Wigilię.
Names of the months, weeks, years, quarters, are in locative:
W maju, w tym tygodniu, w roku 2020, w kwartale.
Sometimes both cases are acceptable, for example with "dzień": w pierwszy dzień miesiąca/w pierwszym dniu miesiąca (on the first day of the month).
However if you mean "in" as the synonym of "within some time", all words turn into accusative: I'll do it within a month = Zrobię to w miesiąc.
Twój tata wyjeżdża w piątek - Your dad going away on Friday
Twój tata odchodzi w piątek - Your dad leaves (your family) on Friday
Twój tata odjeżdża (na tamten świat) - You dad... is going to die on Friday
Twój pociąg odjeżdża ze stacji/z przystanku... - Your train departs...
Twój samolot odlatuje/wylatuje z lotniska w piątek - Your plane departs...
Ehm... what? I can agree with almost everything, but why would anyone say "odjeżdża na tamten świat"? I'd understand "odchodzi" here, sure, but "odjeżdża"? Is he taking a train to heaven?
I'm not sure about odlatuje/wylatuje either, given your odjeżdża/wyjeżdża comparison, shouldn't it also be "samolot odlatuje/twój tata wylatuje"?
I'll think this sentence through as "wyjeżdża" does indeed sound better to me here.