jmc

Since $AB\parallel YZ,$ we know that $\angle A=\angle Z$ and $\angle B=\angle Y.$ That works out nicely, since that means $△ABQ\sim ZYQ.$ If $BQ=12$ and $QY=24,$ that means the ratio of sides in $ABQ$ to $ZYQ$ is $1:2.$

Since $AZ=42=AQ+QZ$ and $AQ=\frac{QZ}{2},$ that means $\frac{3\cdot QZ}{2}=42$ and thus $QZ=\boxed{28}.$