With the help of El Bachraoui's lemma, the creative microscoping method, and a new form of the Chinese remainder theorem for coprime polynomials, we prove a $q$-supercongruence for double series and a $q$-supercongruence for triple series modulo the sixth power of a cyclotomic polynomial. As conclusions, two corresponding supercongruences for double and triple series, which are associated with the (D.2) supercongruence of Van Hamme, are given.