Nestled in the very center of the City-center (the 5th District or Belvaros), there lies a gated vestpocket park with meticulous landscaping, cascading fountains and imaginative playgrounds. 

Originally the private garden of Count Antal Károlyi, this park sits behind the 18th Century mansion that is now the city Petöfi Literature Museum.  One block from the National Museum and two blocks from the two major Metro lines (the Red line at Astoria and the Blue Line at Kalvin ter), the park is also just two blocks from the Great Synagogue (the largest synagogue in the world), three blocks from the Danube, 2 blocks from the historic pedestrian street Vaci utca, and conveniently midway between our two favorite Budapest cafes...the Cafe Central and the Alibi (and three blocks from our favorite pastry shop, the Jeg Bufé). 

Here is what Frommer’s has to say about Károlyi Kert:

This beautifully maintained neighborhood park has benches in the shade, swings, a miniature soccer field, and a lovely fountain with begonias growing around it. The park is filled with the sounds of children all day long.

Then there’s this from http://budapest-tourist-guide.com: 

Károlyi Kert (Károlyi Garden) - perhaps the most beautiful small park in Budapest:  This charming park lies behind the Classicist Károlyi Palace (Károlyi utca 16., off Múzeum körút, V. district, M3 (blue) metro line, Ferenciek tere or Kávin tér station.  The garden is a home to some rare plants like the honey locust, Caucasian wingnut, silver poplars, horse chestnuts. These stately trees were probably planted at the beginning of the 19th century. The Károlyi Palace provides shelter from the traffic and noise of downtown Budapest that's why the park is and ideal place to relax after a walk in the city centre.  Tip: the palace turns into a cultural centre during summer hosting the Summer Evenings in the Károlyi Garden event with classical music, dance, folk and other cool programs (26th June-20th July 2007).

and this from Frommer about its playground:

Tucked away throughout Budapest you will find some charming playgrounds. ... Károly kert, a wonderful little enclosed park in the southern half of the Inner City, is bordered by Ferenczy István utca, Magyar utca, and Henszlmann Imre utca. Enter the park through a wrought-iron gate. Once inside, you'll find swings and seesaws, an enclosed miniature soccer field, a sandbox with a slide, and a nice stretch of green grass to run on. In the middle of all this is a fountain surrounded by flowers. The equipment is not as modern as what you'll find at some of the city's other playgrounds, but the park has a distinct old-world charm. Indeed, it once belonged to the adjacent Károlyi mansion, which was the home of Mihály Károlyi, who served briefly as Hungarian prime minister in 1918. The mansion was recently restored to its old splendor and functions as the Museum of Hungarian Literature (tel. 1/317-3611, ext. 203). Its location in the Inner City makes it a convenient destination. The museum is open Tuesday to Sunday from 10am to 6pm.     

For a history of the palace for which the garden was built, see the website of the Petöfi Literature Museum: http://www.pim.hu/object.EDA790EA-DFC5-4869-B8E6-B6814B65E9DE.ivy 

There is no lovelier, more tranquil, and more secluded spot in any of the world’s major capitals.