Playgrounds in Budapest near sights

Family holidays are great bonding events, but young children have very different needs, and even the most interested junior sightseers can have "had enough" well before the adults. Some time at a nearby playground with one parent can be just ticket, to allow everyone to enjoy the day! Thankfully Budapest has a wealth of great playgrounds many of them perfectly situated for this purpose:

Buda Castle
This small playground, originally built for the local school, is fantastically situated just 20 metres from the Fishermens' Bastion! This and the clean pay-toilets next door are the advantages. The downside is that the playground equipment is quite limited, and as this area is built into the castle hillside, there is an unhealthy drop on the other side of the 1.2m high castle wall; it is useful for a few minutes' breather, but kids definitely have to be supervised.
Sights nearby: Matthias Church, Fishermens' Bastion, Royal Palace, Budapest History Museum, "Hospital in the Rock." 
 Facilities nearby: Toilets, ice cream stalls, supermarket

Hild Square
(Hild tér) (pron. Heeld tair)
This is 300m away from largest square of Budapest, Elisabeth Square, but has a far better playground, with a range of  great toys and climbing equipment, rubber matting, a good sandpit and plenty of shade in the heat of summer.
Nearby sights: Váci street, the chain bridge, pest riverside stroll, St. Stephen's Basillica, synagogue
Facilities nearby: Supermarket (No Toilets)

Károlyi Gardens
(Karolyi kert) (pron. Ka:royee kerth)
This neat and tidy park is one the oldest in the city and has pretty flower beds, and pram friendly pathway and separate play areas for younger and older children. The park stays quite shady in the heat of summer.
Close to: Hungarian National Museum, Great Market hall, Pest riverside walk.
Facilities nearby: Toilets, Coffee shop 30m away, Mini-mart on the corner

Gellért hill slides
This a small, partially hidden area on the side of Gellért Hill. It can be quite hard to find, but if one follows the path uphill from the Cave church to the lookout point, the path for the playground branches off to the left about halfway. It is a collection of slides and tubes  that is great for the toddlers and fun for children up to about 10 years of age.
Close to: Gellért hill viewpoint: Citadella, underground church, Great Market hall.
Facilities nearby: Nothing nearby. The park is quite large and on a steep hill.

Heroes' Square
(Hősök tere) (pron. Hur shirk terre)
 
This epic playground stretches over a wide area and has facilities for older children too, such as ping pong tables and good climbing bars. It is part of a larger park, and is unfenced, so parental vigilance is required. 
Facilities nearby: No toilets nearby (except in the Fine Arts Museum).
 

Other playgrounds worth noting:

Hajógyári Island  (huyodya:ree sigget)  long slides
Far out of centre, but great rest stop if cycling to Szentendre.

Tabán Hillside
This is a very simple set of swings and springy motorbikes, which hardly qualifies as a playground. However, for smaller children, say 2-4, this will be enough. The huge advantage here for the parents is the location. It is at the top of a steep grassy hill, with an absolutely splendid view of the Royal Palace, Elisabeth bridge, the Danube and the Citadella. If you have a ball, or frisbee even larger kids will amuse themselves for a time, struggling up and down the steep hill. This is an open park though and quite a large one, so extra parental vigilance is required. 
Sights nearby: Royal palace (take elevator from Dozsa György square) Buda Castle, Budapest History Museum, "Hospital in the Rock", Rác and Rudás Thermal Baths.

Indoor (Paid) Playgrounds:

Mammut shopping Centre, Mammut 2 or "Kalandpark" LurdyHáz Shopping Mall

These two shopping malls provide an expensive option: paid indoor playhouse in a mall, where you can check your kids in, receive a code without which they will not be released, and then wave goodbye.  Sounds scary? It isn't. This place is a 3-story jungle of slides, bouncy castles and rubber cushions, all netted and padded for safety. For children aged 4-10, it's heaven! They are normally flushed and exhausted after just an hour, when they finally emerge( or more likely when you fetch them out to avoid paying for the next half-hour.)

All of these locations (except the Mammut and LurdyHáz playgrounds) are free, but also outdoors, so do dress appropriately and keep an eye on the weather. Unlike some Southern-European cities, Hungarian playgrounds do not usually have coffee-shops (or any shops) nearby, so bringing your own drinks and snacks are a good idea. On the other hand, sandpits normally have a water fountain nearby which you will see local children safely drinking from. As with any international travel, it may be best to avoid your children drinking the local water if you they are unused to it.  Toilet facilities are not always present or open.