Answer: Street parking is free on the weekend in Budapest as long as you stay in a designated space. (look for spaces at the side of the roads with blue parking meters, then read the meters to be sure).
Anywhere after 6PM Friday Night to 8AM Sunday morning should be free for you around Rákóczi út.
Honestly I think you should easily find a parking space somewhere near where you are staying simply by driving there and looking around at the street parking. You are best to to turn off onto the single lane roads and keep going round the small streets until you find a spot.
If you do struggle to find an exact spot, you could try this street - near where you are staying and has street parking on both sides of the road: https://goo.gl/maps/uqsziWVwkyt
If you still can't find a space, or wish to park in a 24 hour garage, there is one in a shopping center close to where you are staying that costs around 4000HUF (12 EUR) per day: https://goo.gl/maps/U6nzHEstbU32
My advice would be to just park your car near where you are staying and then use public transport to get around.
Hopefully this reaches you in time, good luck!