What Is the Difference between Parseval's Theorem and Plancherel Theorem?
According to the links you shared, and using Wikipedia's notation, Parseval assumes $A(x),B(x)$ are two square integrable (with respect to the Lebesgue measure), complex-valued functions on $\mathbb {R}$  of period $ 2\pi $  with Fourier series

$$A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$
and
$$B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}$$
Then 
$$\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, \mathrm{d}x$$
Plancherel provides a continuous version of Parsevals theorem (in a sense that $a_n$ and $b_n$ are now the $f(x)$ and $g(x)$, respectively and $A(x),B(x)$ are replaced by $ \widehat{f}(\xi), \widehat{g}(\xi)$)