How to find mountain parking in Switzerland?

It depends on the location and the popularity of the area. A good hiking book would tell you, and large parking areas may be shown on car maps and/or openstreetmap. But sometimes you will just park in a bend in the road, and those small parking areas will not be mapped. Google Streetview may help you there.

Switzerland is a country with 62000 km of hiking trails, including 21000 km mountain hiking trails. There are thousands and thousands of places where those cross roads/streets or briefly run along them. There is no generic answer to “can I park here?”, and in some cases the only way to find out is to check it out.

Where a trail simply crosses a road, it may be difficult to park, but Google Streetview may help to inform you. Where it passes through a village or town, you can park in the town, which may or may not be paid parking. In more popular areas, there may be dedicated parking areas at popular starting points. This may or may not be paid parking. Switzerland is a highly decentralised country and it's all up to local authorities. A good hiking book will tell you about parking opportunities, but information about paid/unpaid can quickly be out of date. I don't think anywhere needs booking in advance, except possibly in Täsch, where fools who forgot to leave their car at home park to go to the highly popular destination of Zermatt.

Most trails in Switzerland are accessible by public transportation. Quite a few places are exclusively accessible by public transportation (in this case, the place where you transfer from your car will have parking). Whether or not it is reasonable to get a car at all depends very much on where you're going. The northern parts of the valleys of Ticino, probably yes. A popular hiking area like Zermatt, certainly not ­— cars are prohibited there.