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:$$\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86\pi^{2}}{1104}\right]\frac{1}{4096^{k}}=0,$$$$\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555\zeta(3)}{77\pi}-\frac{32G}{11},$$ where $\mathsf H^{(r)}_k:= \sum_{0<n\leq k}\frac{1}{n^r}$, $L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots$, $G:= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}$, and $\zeta(3):= \sum_{n=1}^\infty\frac1{n^3}$.