So you can't go buy a bucket? Use the bathtub? How old are you?

Time to do things by hand

Calm the fuck down, Cinderella. Get a plumber to fix the sink and make sure you give your parents the bill.

My father's "kitchen" didn't have a sink, so they did all the dishes in the bathroom sink. Technically he was living in an old converted mink farm minus the cages, so they only needed water in one room. Odd, but it worked.

just wash them in the bathroom sink or tub, bonus, fill the tub with soap and water, wash them all then when you're done drain it and spray it down with the shower to rinse them! get a dish towel and and a box and dry them as you put them in the box then go put them away! actually if its just you at home there shouldn't be too many dishes anyway. also contact your parents about the sink and ask them if there's anyone they'd like you to call to fix it.

i can only hope this was a prank

I'm sure there are other sinks in the house.

Do you live in a TV show? I can't imagine any real-world scenario where this would actually happen. One "absolutely imperative" chore; apparently only one appliance available to complete said chore, which conveniently breaks; and a dim-witted protagonist (most likely a teenage boy or a man-child) who doesn't have the necessary gray matter to work out that there are other solutions to this problem. Also, if you and your parents live in a property with a single sink in the whole place, I think clean dishes should be a little further down the list of "Shit I have to deal with".