Correlation coefficients play a pivotal role in quantifying linear relationships between random variables. Yet, their application to time series data is very challenging due to temporal dependencies. This paper introduces a novel approach to estimate the statistical significance of correlation coefficients in time series data, addressing the limitations of traditional methods based on the concept of effective degrees of freedom (or effective sample size, ESS). These effective degrees of freedom represent the independent sample size that would yield comparable test statistics under the assumption of no temporal correlation. We propose to assume a parametric Gaussian form for the autocorrelation function. We show that this assumption, motivated by a Laplace approximation, enables a simple estimator of the ESS that depends only on the temporal derivatives of the time series. Through numerical experiments, we show that the proposed approach yields accurate statistics while significantly reducing computational complexity, from $O$($n$ log $n$) to $O$($n$). In addition, we evaluate the adequacy of our approach on real physiological signals, for assessing the connectivity measures in electrophysiology and detecting correlated arm movements in motion capture data. Our methodology provides a simple tool for researchers working with time series data, enabling robust hypothesis testing in the presence of temporal dependencies.