Learn Spanish (Spain): Mi suegro es muy simpático.

No, this is a classic false friend! In Spanish, simpático means "nice", "friendly", or "pleasant". If you want to say someone is sympathetic or understanding, you would use a word like comprensivo.

Why do we use es instead of está here?

We use es (from the verb ser) because being nice (simpático) is seen as a core part of his personality. We use ser for permanent personal characteristics and traits, whereas estar is used for temporary moods or physical states.

How would I say "mother-in-law" or "parents-in-law"?

You just change the ending! Your mother-in-law is your suegra, and your parents-in-law (together) are your suegros. Spanish groups mixed-gender family members under the masculine plural form.

How would this entire sentence change if I were talking about my mother-in-law?

You would need to change both the noun and the adjective to their feminine forms. The sentence would become: Mi suegra es muy simpática. The word mi (my) and muy (very) do not change for gender.