Parabolic fiting for sub-sample precision
You can use the formulas presented in the answers to: How to calculate a delay (correlation peak) between two signals with a precision smaller than the sampling period?

To recap, find the largest value, called $\beta$. Take also the values of the samples just to the left of it, $\alpha$, and just to the right of it, $\gamma$. Then calculate the peak position $p$ of Lagrange parabolic interpolation by:


  $$p = \frac{1}{2} \frac{\alpha - \gamma}{\alpha - 2\beta + \gamma}$$


If your signals are white you can get an unbiased estimate $d$ of the peak position from the biased estimate $p$ by:


  $$d = \frac{\sqrt{32p^2 + 1} - 1}{8p}$$


These estimates give the peak position relative to the position of the sample with the highest value.