I would tend to think that the multifunction switch was bad. It has often failed for many reasons in the past, would it be possible to disconnect the MF switch and mimic it with powered jumpers?
Used ones are not cheap and not so easy for some to replace.
If you take the MF switch apart, that’s if it is possible, the park contacts may be burnt out. Is it always the same number of cycles?
If it does not park correctly then there is always power to operate the wipers, the park switch is operated every cycle but is over ridden by another switch which has power or no power as dictated by the MF switch.
I looked at the insides of a wiper motor once years ago and the park or continue to cycle mechanism was so simple, it was the subject of a lawsuit that went on for ages, the inventor won millions back in the sixties.