I think you're right and it would sound better with "is coming" but I also think this translation just sounds awkward in general. I believe it's more common to say something like "Dad cooks dinner before my brother gets/comes home"
I don't understand while there are two tenses in the translation. I'm not an English native speaker, but for me it would be more logical as follows:
Dad is cooking dinner while my brother is going home. Dad cooks dinner while my brother goes home.
I think general consensus here is that the translation given is kinda crummy. I think "Dad cooks dinner while my brother walks home" or "is on his way home" is probably the closest natural-sounding translation, in essence.
This should definitely be something like, "Dad cooks/is cooking dinner while my brother is on his way home." The current translation is so clunky I don't believe any native English speaker would ever say this.