In order to use Pearson's correlation to measure the similarity of two time series, is normal distribution of both time series a necessary condition?
Several questions are bundled together here:
Pearson correlation measures linearity of relationship, not similarity of values. $y$ and $a + by$ have correlation $+1$ for $b > 0$: make $a$ as different from zero or $b$ as large as you like, and the similarity is at best one of shape, not size. (Concordance correlation does measure agreement in the sense that $y = x$ is required for perfect positive correlation.)
Normality of distribution is not a requirement to measure correlation; correlation is perfectly well defined as a descriptive statistic (or even as a non-statistical property; it's just a cosine from one point of view) so long as both variables are genuinely variable. Marginal distribution is of concern if you wish to test for significance, e.g. produce a P-value.
But you have time series and can expect dependence in time and (quite likely) other kinds of structure. The standard machinery for Pearson correlation is for independent data, and no inference for Pearson correlation for time series can be taken seriously without adjustment for dependence structure.
What is the real problem? For assessing similarity of two time series, I would always start with plotting the series and examining (as appropriate) the difference or ratio between them. The next step is harder and entails modelling the series to see if they have the same structure; others will predictably make positive suggestions here. Also, here as elsewhere, a single summary measure will rarely do justice to the fine structure of interesting data.