This question also discusses what I am asking but the answer states that Parseval's theorem is for periodic functions as per wikipedia's notation here.
However, Parseval's relationship also holds for continuous-time aperiodic signals as follows:
$$\int_{-\infty}^{\infty}|x(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\mathrm{X}(j\omega)|^2 d\omega$$
so then what difference remains between the two other than the $2\pi$ correction and how should they be interpreted differently?
In most (engineering) literature, the two names are used interchangeably, with Parseval's Theorem being used more often. WolframMathWorld agrees with the view that Plancherel's Theorem is used for non-periodic continuous functions (i.e., using the Fourier transform), whereas Parseval's Theorem is used for Fourier series (i.e., for periodic functions). However, many authors don't seem to adhere to this convention.
Another distinction I've seen used is that the preservation of energy is called Plancherel's Theorem, whereas the preservation of inner products
$$\int_{-\infty}^{\infty}x(t)y^*(t)dt=\int_{-\infty}^{\infty}X(f)Y^*(f)df$$
is called Parseval's Theorem (cf. [1]). However, the author of [1] then suggests
We shall use "Parseval's Theorem" for both.
Sometimes you also see the term Parseval-Plancherel identity (e.g., here), which doesn't help in clearing up the confusion.
In sum, more often than not Plancherel's Theorem and Parseval's Theorem are used to mean the same thing, namely preservation of energy and preservation of inner products for Fourier series as well as Fourier transforms (in the $L_2$ sense).
Note that this answer represents an engineering point of view, as seems appropriate for this site. But even in some of the more mathematical literature I couldn't find a consequent distinction between the two names.
[1] A.Lapidoth, A Foundation in Digital Communication, p. 72
