I didn't have time to write out an answer last night, so I'll expand on my comments here. The goal is to show that orthobasis expansions are geometry preserving.
Let $\{\phi_{\gamma}\}_{\gamma \in \Gamma}$ be an orthobasis for a signal in space $G$. For any two signals $x,y \in G$ we have the synthesis operator
\begin{equation}
\Phi[\{\alpha(\gamma)\}_{\gamma \in \Gamma}] = \sum_{\gamma \in \Gamma}\alpha(\gamma)\phi_{\gamma}(t)
\end{equation}
and the analysis operator
\begin{equation}
\Phi^{*}[x(t)] = \{\langle x(t),\phi_{\gamma}(t) \rangle \}_{\gamma \in \Gamma} = \{\alpha(\gamma)\}_{\gamma \in \Gamma}
\end{equation}
We wish to show that for any orthobasis
\begin{equation}
\langle x(t),y(t) \rangle = \sum_{\gamma \in \Gamma}\alpha(\gamma)\bar{\beta}(\gamma)
\end{equation}
where
\begin{equation}
\alpha(\gamma) = \langle x(t),\phi_{\gamma}(t) \rangle, \; \beta(\gamma) = \langle y(t),\phi_{\gamma}(t) \rangle
\end{equation}
Explicitly writing this out, we get
\begin{aligned}
\langle x(t),y(t) \rangle &= \langle \sum_{\gamma \in \Gamma}\alpha(\gamma)\phi_{\gamma}(t),\sum_{\gamma ' \in \Gamma}\beta(\gamma ')\phi_{\gamma '}(t) \rangle \\ &= \sum_{\gamma}\sum_{\gamma '}\alpha(\gamma)\bar{\beta}(\gamma ') \langle \phi_{\gamma}(t),\phi_{\gamma '}(t)\rangle
\end{aligned}
Using the definition of an orthobasis
\begin{equation}
\langle \phi_{\gamma}(t),\phi_{\gamma '}(t) \rangle =
\begin{cases}
1, & \gamma = \gamma ' \\
0, & \gamma \neq \gamma '
\end{cases}
\end{equation}
we get
\begin{equation}
\langle x(t),y(t) \rangle = \sum_{\gamma}\alpha(\gamma)\bar{\beta}(\gamma)
\end{equation}
The energy in a signal x(t) is then
\begin{aligned}
\lvert\lvert x(t)\rvert\rvert_{2} &= \int \lvert x(t) \rvert^{2}dt \\ &= \int x(t) \bar{x}(t) dt \\ &= \langle x(t),x(t) \rangle \\ &= \sum_{\gamma}\alpha(\gamma)\bar{\alpha}(\gamma) \\ &= \sum_{\gamma} \lvert \alpha(\gamma) \rvert^{2}
\end{aligned}
For the Fourier transform, we get
\begin{equation}
\{\alpha(\omega)\}_{\omega \in \Omega} = \{\langle x(t),\phi_{\omega}(t)\rangle\}_{\omega \in \Omega}
\end{equation}
where
\begin{equation}
\phi_{\omega}(t) = e^{-j\omega t}
\end{equation}
Thus, Parseval's theorem with respect to the Fourier transform being energy preserving is a consequence of the more general conclusion that orthobasis expansions are geometry preserving.
1 Justin Romberg's lecture notes on compressed sensing - Lecture 1
