As in years past, you can bike out to the Sziget festival (August 13-18 at Hajogyari Island) and leave your bike in a free-of-charge, attended parking facility. In addition to secure parking, you can also get minor repairs and maintenance done. Check out the blurb on the English-language page at http://www.sziget.hu/festival_english/travel
For those who've never biked out there, the Buda entrance at the Filatorigat HÉV stop is along a designated bike route. Get on the Budai rakpart path and follow the signs. After you pass the Aquincum Hotel, the path diverges from the HÉV tracks at Arpad hid and goes on cobblestones through Fô tér at Óbuda and the housing estate immediately to the north. After that, you're directed back toward the HÉV tracks and you'll soon see the massing crowds of Sziget revelers.
Sorry to say, I have no idea if there's an easy way to penetrate the crowd or whether you just have to line up with everyone else and find the bike park after you gain admittance. Presumably, you can get these directions when you finally get the head of the line. Best of luck!