This is one of 24 "Tajchy": a unesco heritage preserved artificial water reservoirs made in 18th century for mining purposes. Now used for recreacional purposes (swimming). I highly advise to pay some of them a visit, you can find some campspots (and freecamp too) along some of them. Perfect for cycling around in the hills.
More info on Tajchy:
Good resources on Tajchy, cycling routes and accommodation:
Piece of grassland next to river. You could also camp on the other side of the river, there are some picknick tables and a small, slow running well. Closeby runs the train over the bridge, but it is a very small one and stops at night. A few hundred meters north you could camp under big trees near the football field. Local people very friendly.
A huge paved/grassy parking lot, lots of leveled space. Could get crowded during the summer festivals or end of the week if the weather's nice, but there's a lot of space.
Just next to the park(Sad Janka Kráľa), across the park there's a huge shopping mall with anything you need.
Also right next to the "Old Bridge", which is the newest bridge in Bratislava - the city center begins just across the bridge, so 10-15 minutes walking and you're there.
Free, no amenities.