Counterfactual explanations have been successfully applied to create human interpretable explanations for various black-box models. They are handy for tasks in the image domain, where the quality of the explanations benefits from recent advances in generative models. Although counterfactual explanations have been widely applied to classification models, their application to regression tasks remains underexplored. We present two methods to create counterfactual explanations for image regression tasks using diffusion-based generative models to address challenges in sparsity and quality: 1) one based on a Denoising Diffusion Probabilistic Model that operates directly in pixel-space and 2) another based on a Diffusion Autoencoder operating in latent space. Both produce realistic, semantic, and smooth counterfactuals on CelebA-HQ and a synthetic data set, providing easily interpretable insights into the decision-making process of the regression model and reveal spurious correlations. We find that for regression counterfactuals, changes in features depend on the region of the predicted value. Large semantic changes are needed for significant changes in predicted values, making it harder to find sparse counterfactuals than with classifiers. Moreover, pixel space counterfactuals are more sparse while latent space counterfactuals are of higher quality and allow bigger semantic changes.