You really don't need heat during summer for your roaches, they will breed regardless.
My roaches are breeding with no heating pads.
It will also save your parents on power, but if you want your roaches to breed in winter they must be heated.(which you all ready know)
I get where your parents are coming from.
Things can go wrong, with anything using electricity.
I now heat cable is a bit more on the dangerous side, there has been a couple incidents where heat cable has started a fire. one where a guys collection of snakes caught on fire.
So it can happen, not saying that it will.
I have never heard of heating mats setting alight, or melting plastic.
I'm going to give you some solution and suggestion. Not that any thing will go wrong, but there is a couple of things you can do, for prevention, which in turn might put your parents at ease.
As Usumbara said get a wall mounted heater, I find this is much better in winter, as it keeps the roaches nice and toasty in the enclosure and you can heat a couple of container at once and the room your spider are in, the only thing with using a wall mounted heater is that it does use a bit more power than your heating pads do.
You can also put a timer on your roaches with a thermostat to control the heat. this also brings down the risk of your heating pads getting to hot, you can also put your heating pads off for 6 hours and on for the rest of the day.
So here is a solution, If in the unlikely case that your heating pad sets alight.
To prevent a fire from spreading, make sure the roach container is no where near curtains, on carpets, wood table or wood shelving, make sure there is nothing lying around the container that can burn.
So the garage will be the perfect place, a room that has tiles, if you don't have that and they are being kept in your bed room go buy yourself some tiles from a Tile company and pack them on the floor so it is wider than your container and bob is your uncle.
Or take the mate to and electrical company and talk to an electrician or go to your local fire department, they will give you an honest answer.