  Two $q$-supercongruences of truncated basic hypergeometric series containing
two free parameters are established by employing specific identities for basic
hypergeometric series. The results partly extend two $q$-supercongruences that
were earlier conjectured by the same authors and involve $q$-supercongruences
modulo the square and the cube of a cyclotomic polynomial. One of the newly
proved $q$-supercongruences is even conjectured to hold modulo the fourth power
of a cyclotomic polynomial.
