Understanding Parseval's Theorem with Discrete Wavelet Transform
Parseval's identity and Plancherel's theorem finally boil down to orthogonality. When one decomposes a data (with samples), via a scalar product, onto an  orthogonal sequence (yielding coefficients), there exists a certain preservation (equality, up to a proportionality factor) of energy between samples and coefficients. There are some technical conditions, and in certain cases, one only get inequalities (cf. Bessel's inequality) or frame bounds.
The equation for the discrete wavelet transform (DWT) might be incomplete, with respect to the indices. I for instance think that in the second term of the RHS, the scaling factor should be $N_j$, not $N_J$ (and this depends a bit on how discrete wavelets are implemented). Basically, an orthogonal wavelet transform projects data onto basis elements gathered in groups called subbands. Each wavelet subband comes from $N_j$ vectors, with an additional $N_J$ vectors for the approximation. And normally, the total number of vectors  should be (about, honestly, this depends on signal extension) the number of samples $N$, in other words: $N=N_J +\sum_{j=1}^JN_j$.