A regular epimorphism is a morphism that is the coequalizer of some parallel pair. Every regular epimorphism is a strong epimorphism.
The coequalizer universal property supplies the diagonal filler against any monomorphism, so every regular epimorphism is strong. In a regular category, every strong epimorphism is regular: factor it as a regular epimorphism followed by a monomorphism and use the lifting property to make the monomorphism an isomorphism.

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