We study a one-parameter family of polynomial $J$-fractions whose coefficients depend polynomially on the index and on an integer parameter $u\ge 0$. After a factorial normalization of the denominator sequence, the difference of two consecutive convergents factors into the fixed central-binomial Apéry-type term $1/(n^2(\genfrac{}{}{0ex}{}{2n}{n}))$ and a rational factor determined by a normalized polynomial family $P_u$. We construct $P_u$ by an explicit parameter-raising operator. We then prove a parameter-shift telescoping identity, which allows induction on $u$ and gives

$$X(u)={(\genfrac{}{}{0ex}{}{2u}{u})}^3\frac{{\pi }^2}{18}+{\rho }_u,{\rho }_u\in \mathbb{Q}.$$

Thus, the operator identity and the telescoping identity provide the algebraic mechanism behind the evaluation of the whole family.