I don't think there is any centralised database of parking fees for the whole of Switzerland, so you'd effectively have to look up each place. Parking fees are often required even for roadside parking and the rules and fees vary by town. However it may pay off to have a closer look at the fine print on the parking meter, often times overnight parking or weekends might be free.
However, there is one possibility: If you are looking to park just for a short while you can look out for blue parking lots which can be used with a blue parking disc. The disc can be bought at any petrol station or post office. The rules are not exactly the same as in neighbouring countries:
- It allows for 1 hour free parking Monday to Saturday from 08:00-11:30 and from 13:30-18:00. However you are allowed to put the mark on the next half hour (you arrive at 09:00, you can put it on 09:30) so this can effectively give you 1.5 hours.
- If you arrive during lunch, you can stay until 14:30, if you arrive at or after 18:00 you can stay until 09:00 the next morning.
Note that it will be hard to find those lots in big cities such a Zurich, which are also the places where parking will be very expensive. In those cases it may be cheaper and more convenient to park outside the city and take the train/tram/bus into the centre.
Just to add: There are also often white parking fields which are free as long as there is no sign indicating that they are private or payable. However, as mentioned above, you are unlikely to encounter these in bigger cities.