Washed should be correct, but "was washing" is also correct.

My father was washing...

I not very good with English grammar, sorry, but I can just say what sounds right to me. Either „My father was washing/washed his car while my mother and I were preparing..(what? dinner? The sentence is incomplete). Either one sounds fine to me. I really don’t know the difference between past continuous and simple past in English verb tenses. German is so much easier in this respect.

was washing

was washing