Taking pictures in Budapest’ spas is not regulated, so, generally, yes, you can bring your camera, mobile phone with you and feel free to take pictures and videos at the world famous spa complexes, and upload them on social sites, making your friends envy.
However, please bear in mind, that others may also take photos of you, while soaking in the hot water. According to the general etiquette, it is not considered to be polite if you take upfront body photos of half naked people without their expressed permission, but it is accepted to take panoramic photos in the baths and pools. Please respect the privacy of others if they do not want to be on a picture or video.
As belongings and valuables are not insured via the bath entry ticket, please leave your brand new and high-tech camera or video equipment at home, especially if there is no one to take care of them while you are in the pools. There are no security cameras observing the pools either. It is your responsibility to look after yourself and your belongings. Of course, you can use the locker and the cabin for safe storage indoors, but accidents do happen as people sometimes forget to lock their lockers.