No question that a car offers flexibility. And, if you have a large group, rail costs can add up.
That said, if you have not experienced public transport in Switzerland, you may be surprised by the frequency of service.
In this case, public transport runs every 30 minutes from "Pilatus Zentrum" , does not have to go back through Luzern, and takes 40 minutes. Take a look at the attached screen grab of a random date in May from the timetable. You do have an 11 minute walk in Kriens from "PB" to "Pilatus Zentrum" stop.
Individual tickets cost CHF 4.30 with a half fare card, probably twice that with a full fare, although there are various day-card and 24 hour passes that could be useful if you plan to travel further in or near Luzern that day.
which is from mapsearch.ch with some transport overlays turned on in the menus shows that the nearest parking facility is at Parking Garage Hoffmat, which is about the same walking distance as the bus stop at Kriens, Zentrum Pilatus.
There is a bus stop right at Pilatus Bahn (PB) Kriens, but I did not explore its routes or schedules. I let the timetable pick the route in the screen grab attached.
If you are feeling energetic, the train station is a 2 km. walk from the Talstation (valley station) of the cableway.
A screen grab of the satellite view does show limited parking at Kriens PB...attached. I have no idea of cost, or whether the parking area is associated with the cableway, but such lots often have a central parking meter (Zentral Parkuhr.)