Yes the non-used insurance premium is even paid back - IF its not in the first year of the contract. There is this exeption in the law in order to cover the setup-fees from the insurance but after the first year, you are good to go… Fun fact: This basically applies to almost every insurance based on the VVG (Versicherungsvertragsgesetzt) so you can even cancel your additional health insurance after getting a massage or a fitness membership
So considering the parking damages you need to know the following:
If you have a special coverage for parking damages, check the covered amount (there a versions like: unlimited, 1k, 2k, 3k) and check the deductible of that coverage (ranges from 0.- to a few k as well)
If you don’t have a parking damage coverage included, but still have full-kasko, it will be covered after the kasko-deductible which is mostly 1k or CHF 500.-
If you don’t have full-kasko but a partial kasko, it might work as well but then you need to find a damage which is covered, for example a crack in your windshield etc.
If 1-3 is checked and you have a damage where the damage-value is above the deductible, you can go ahead and then your question triggering a premiumraise with the new insurance is relevant:
4.1. If you didn’t had any claim of any kind with your car insurance in the last 3-5 years the impact might be low to zero, since most insurances ignore 1 claim.
4.2. If you had more than one claim in the past 3-5 years, you should be more cautious and get quotes where you indicate the claims before you declared them to the old insurance.
4.3 If you have a parking-damage coverage now and won’t get it with the new insurance, the claimhistory won’t matter…
And now something which very few persons know but is a real mustachian move: When you have a claim with any insurance, specially with cars → you are not obliged to have the repair done! That means that if you have a parking damage which would cost 2k to fix, you can decided to have it fixed, or to take the money (they only deduce the VAT). So since I don’t give a damn about the exteriors of my car, I once decided to have a big hail damage fully paid out (8.8k) → You have to inform them while declaring the claim, that you will decide later and want first them to estimate the damage-amount, then no repair will be done before your decision.
If you downgrade your insurance, do what you want but I still strongly encourage you to keep the coverage for gross negligance (grobfahrlässigkeit) - its very cheap but could save you seven digits in a worst-case scenario.
The cheapest car insurances at the moment, based on my experience are the following and you can in most cases get a quote online:
Dextra.ch, Smile-direct.ch, automate.ch, Allianz Suisse (no online offering) and a white label offer from Postfinance: PostFinance Autoversicherung | PostFinance
I suggest you check them out and conclude only yearly contracts, so that you can stay flexible.
I hope this helps,