Did you try some plots? Show us. And, partial correlations correspond to correlations of residuals, see https://en.wikipedia.org/wiki/Partial_correlation. So, the corresponding plots woud be plots of residuals, which very well could be helpful.

More details. With variables $x,y,z$ we want to "partial out" $z$. Then estimate the regressions $$ x=\alpha_0+\alpha_1 z + \epsilon_x \\ y=\beta_0 +\beta_1 z + \epsilon_y $$ with residuals $\hat{x}=x-\hat{\alpha_0} -\hat{\alpha_1} z$ and $ \hat{y}=y-\hat{\beta_0} -\hat{\beta_1} z$. Then the partial correlation is $$ \text{cor}(\hat{x}, \hat{y} ) $$ and you can make a plot of this residuals, which will show any nonlinearities. All of this still makes sense if the linear regressions above is replaced by nonlinear regressions.

Did you try some plots? Show us. And, partial correlations correspond to correlations of residuals, see https://en.wikipedia.org/wiki/Partial_correlation. So, the corresponding plots would be plots of residuals, which very well could be helpful.

More details. With variables $x,y,z$ we want to "partial out" $z$. Then estimate the regressions $$ x=\alpha_0+\alpha_1 z + \epsilon_x \\ y=\beta_0 +\beta_1 z + \epsilon_y $$ with residuals $\hat{x}=x-\hat{\alpha_0} -\hat{\alpha_1} z$ and $ \hat{y}=y-\hat{\beta_0} -\hat{\beta_1} z$. Then the partial correlation is $$ \text{cor}(\hat{x}, \hat{y} ) $$ and you can make a plot of this residuals, which will show any nonlinearities. All of this still makes sense if the linear regressions above is replaced by nonlinear regressions.

For an example see the video: https://www.youtube.com/watch?v=gLenrMgrlc8