I would rather you washed it yourself than (you) let your mother do it.
I would rather he washed it himself than (he) let his mother do it.
is a sentence about a hypothetical situation: the reality is you let/he lets your/his mother wash it, and I wish you/he would not act like this.
That is why it contains the auxiliary "will", meaning – here – something like "be happy to see something happening", in the CONDITIONAL SIMPLE, and the verbs "wash" and "let" in the PAST SIMPLE SUBJUNCTIVE, as in Italian!
The past simple in the subjunctive mood cannot be differentiated from the past simple in the indicative mood, except for the verb "to be", which is "were" for all persons: If I were you, etc. ; If he were me, etc.
Your sentence contains an implied if-clause belonging to the second conditional (unreal present) structure:
conditional simple + if + past simple (subjunctive).
So, if I rephrase your sentence,
I would be happier if you/he washed it yourself than (if you/he) let your mother do it.
I would be happier to see you/him washing it yourself/himself than (you/him) letting your mother do it.