parseval for a continuos but limited signal
Sure. You can can just integrate in the time domain from $0$ to $T$ since the area outside  $[0, T]$ is 0.

Please note that you still must integrate from $-\infty$ to $+\infty$ in the frequency domain since finite support in the time domain implies infinite support in the frequency domain (and vice versa).

Things are different if the function is periodic.