Ideally check first at the wiper motor using a multimeter/voltmeter to see if power from the "on" supply goes off when switched off and just leaves the "park" supply live.
This is just in case something else like a faulty switch, relay or ECU command is causing the issue.
However if when you turn off the ignition it doesn't "park" until ignition goes back on again, then it does point towards the wiper motor in my opinion.
Maybe others on Forum have a different suggestion? Either way I woulds always check all options before spending money as a few minutes with a multimeter should prove or disprove my point.