where is the best place to park to visit the Research Center

There is free parking after 4 pm and on Saturday behind the Church History Museum on the corner of North Temple and West Temple. The entrance to the parking is from North Temple Street.

That's the best answer. If you can find parking near the train line that also might be a good choice as the train stops a block away.

There is limited pay street parking on both sides in front of the Family History Library. Also just a half block north is a pay parking lot.