There is parking at the Swiss Miniature in Melide. I cannot remember if it is free or paying.
As for Lugano, there is public parking which is clearly indicated by signs and this is paying. I would not attempt to drive around the town of Lugano since the centre is pretty pedestrianised so you cannot get in there with your car anyway and the public transport is good - and it is a relatively small place. However if you are planning on going to Morcote, for example, you can certainly drive there.
Here is an except from a very helpful post from "Bruno" in Lugano sometime ago:
In Via Clemente Maraini and in Via Balestra is impossible to park. Parking in the Parking-Houses is expensive in Lugano, more or less as everywhere in Western Europe, although some major European towns would be even more costy. Since 1st July 2010 all parking-fees have been equalized in Lugano; thus, they are all expensive. I suggest 2 parkings-lots: Parking Balestra and Parking Campo Marzio. Cheaper is the Parking House of Paradiso (the next Municipality, just between Lugano-Lake and the highway exhit Lugano-Sud). From Paradiso there is the bus going to Lugano-centre (it takes 10 minutes). Alternatively, you walk from that parking to the boat-pier of Paradiso (10 minutes walking). Have a nice stay.
I would add that the walk from Paradiso to the city centre is very lovely (along the lake) and doesn't take much time!
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