Series involving central binomial coefficients and higher-order harmonic numbers

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:$$\sum _{k=0}^{\mathrm{\infty }}{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^3[{\mathsf{H}}_{2k}^{(2)}-\frac{25}{92}{\mathsf{H}}_k^{(2)}+\frac{735L_{-7}(2)-86{\pi }^2}{1104}]\frac{1}{{4096}^k}=0,$$$$\sum _{k=0}^{\mathrm{\infty }}{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^3[{\mathsf{H}}_{2k}^{(3)}-\frac{43}{352}{\mathsf{H}}_k^{(3)}]\frac{42k+5}{{4096}^k}=\frac{555\zeta (3)}{77\pi }-\frac{32G}{11},$$ where ${\mathsf{H}}_k^{(r)}:={\sum }_{0<n\le k}\frac{1}{n^r}$, $L_{-7}(2):={\sum }_{n=1}^{\mathrm{\infty }}\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^2}-\frac{1}{5^2}-\frac{1}{6^2}+\frac{1}{8^2}+\cdots$, $G:={\sum }_{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1{)}^2}$, and $\zeta (3):={\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.