"There aren't any sweets at my parents'."
Translation:Il n'y a aucune sucrerie chez mes parents.
Not normally in British English, no. Those are desserts, or puddings.
"Sweets" covers all your basic sugary confectionary that you might pick up at the corner shop or newsagent's, such as chocolates, toffees, marshmallows, jelly beans, and Starburst (fruit taffies). These are not normally considered "candy" in the UK, a term which is reserved for hard sweets made from boiled sugar, like candy canes, mint humbugs, lollipops, and Brighton rock.