Learn Finnish: Hän lopettaa viestin sanoilla: parhain terveisin. - He ends the message with the words: best regards.

This is the instructive case, which ends in -in and almost always appears in the plural. It expresses the method or instrument used to do something. In modern Finnish, the adessive case has mostly taken over this function (as seen with sanoilla in this very sentence), but the instructive survives in fixed, idiomatic phrases like parhain terveisin (with best regards) and paljain jaloin (barefoot).

Is parhain related to the word paras (best)?

Yes, here it is the plural instructive form of paras. To build it, you take the plural stem parhai- and add the instructive ending -n. You might also recognize parhain as an alternative, slightly poetic dictionary-form superlative of hyvä (good), but in this phrase, it is strictly an adjective agreeing with the plural instructive noun terveisin.

Why use the adessive sanoilla instead of the illative sanoihin with the verb lopettaa?

Both are correct, but the nuance is slightly different. Using the illative sanoihin is extremely common and treats the words as the endpoint where the message stops. Using the adessive sanoilla treats the words as the instrument or tool used to perform the closing action.

Why is the object viestin in the genitive-looking accusative case instead of the partitive?