The GooFit Framework is designed to perform maximum-likelihood fits for arbitrary functions on various parallel back ends, for example a GPU. We present an extension to GooFit which adds the functionality to perform time-dependent amplitude analyses of pseudoscalar mesons decaying into four pseudoscalar final states. Benchmarks of this functionality show a significant performance increase when utilizing a GPU compared to a CPU. Furthermore, this extension is employed to study the sensitivity on the $D^0 - bar{D}^0$ mixing parameters $x$ and $y$ in a time-dependent amplitude analysis of the decay $D^0 rightarrow K^+pi^-pi^+pi^-$. Studying a sample of 50 000 events and setting the central values to the world average of $x = (0.49 pm0.15) %$ and $y = (0.61 pm0.08) %$, the statistical sensitivities of $x$ and $y$ are determined to be $sigma(x) = 0.019 %$ and $sigma(y) = 0.019 %$.
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