Let $\Phi_n(q)$ be the $n$-th cyclotomic polynomial in $q$. Recently, the author and Zudilin provide a creative microscoping method to prove some $q$-supercongruences mainly modulo $\Phi_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this creative microscoping method to confirm some conjectures on $q$-supercongruences modulo $\Phi_n(q)^2$. We also give some parameter-generalizations of known $q$-supercongruences. For instance, we present further generalizations of a $q$-analogue of a famous supercongruence of Rodriguez-Villegas: $$ \sum_{ k=0}^{ p-1}\frac{ { 2k\choose k}^2}{ 16^k} \equiv (-1)^{ (p-1)/2}\pmod{ p^2}\quad\text{ for any odd prime $p$.} $$
Categories: math.NT math.CO