Hi Cookie Monster,
I'd second what Mr. E said about the hotel. Since you're staying at the Parisi Udvar, the easiest path is honestly just to ask the front desk about their laundry or pressing service when you check in on Monday evening. A hotel of that caliber will absolutely have an in-house service, and they can often arrange express or same-day turnaround if you ask nicely and explain you have family photos coming up. That gets you fresh clothes without hunting around an unfamiliar neighborhood.
If you'd rather do it yourself or the hotel's price feels steep, there are drop-off wash-and-fold services around District VII (the Jewish Quarter area), which is a quick walk or short tram ride from downtown. The reception staff can usually point you to the closest one, and they'd know which offer next-day service.
One tip from my own travels: bring your laundry to the desk the evening you arrive, confirm the turnaround time and the price in writing, and separate anything delicate. It saves a lot of worry.
Enjoy Budapest, it's a fantastic city, and have fun at the family photos!