There are lots of parking spaces all over the village but you may have to look for them. You can even park your car in two countries at the same time if you want and if you know where to do it (at the square behind the former station). Almost a shame that parking is for free, I wonder which country would claim the fee if you'd park too long.
There are buses all day long from Breda, ride was about 30 minutes, I do not remember the price or which bus number, I did kept the ticket, it is somewhere here on my desk, when I will find it I will add more details.
I'm sure there are many ways to get to Baarle Nassau but the one I know about is taking bus #132 from Breda. It leaves every hour from the central bus startion which is right by Breda's train station.
Before taking the bus make sure you have the ticket card which will be stamped by the driver. You can get the card at the vending machine of the Albert Hijn supemarket which is located right at the station. Get the one with 15 stripes (costs 6,40 Euro) it will be enough for the roundtrip. The bus has 2 stops in Baarle Nassau the second one is closer to the city center.