On Children’s Day, May 5th, a lot of zoos, parks and museums throughout Japan are free in for children. Apart from that day, there are other days in Golden Week that some places offer free entry for all visitors. Here is a list of some of the places in the Tokyo area, with the free entry dates specified. These are free in for all visitors unless otherwise stated. The links provided are to the English webpages of official sites, with the exception of Yumenoshima park which is in Japanese.
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NHK Studio Park ＮＨＫスタジオパーク
Free for the whole of Golden Week, April 29th to May 5th.
http://www.nhk.or.jp/studiopark/guide/pamphlet/stp_pamph_en.pdf
Showa Memorial Park 昭和記念公園
Free entry on April 29th, May 5th and May 15th for children under 16.
http://www.showakinen-koen.jp/html/english/
Shinjuku Gyoen National Garden 新宿御苑
Free entry on April 29th, May 4th and May 5th for children under 16.
http://www.env.go.jp/garden/shinjukugyoen/english/index.html
Inokashira Park
Free entry for all visitors on May 4th. Free entry for kids on May 5th.
http://www.tokyo-zoo.net/english/index.html
Tama Zoo
Free entry for all visitors on May 4th. Free entry for kids on May 5th.
http://www.tokyo-zoo.net/english/index.html
Ueno Zoo
Free entry for all visitors on May 4th. Free entry for kids on May 5th.
http://www.tokyo-zoo.net/english/index.html
Tokyo Sea Life Park
Free entry for all visitors on May 4th. Free entry for kids on May 5th.
http://www.tokyo-zoo.net/english/index.html
Gardens of the former residence of Iwasaki 旧岩崎邸庭園
Free entry on May 4th.
http://teien.tokyo-park.or.jp/en/kyu-iwasaki/index.html
Gardens of the former residence of Furukawa 旧古河庭園
Free entry on May 4th.
http://teien.tokyo-park.or.jp/en/kyu-furukawa/outline.html
Kiyosumi Gardens 清澄庭園
Free entry on May 4th.
http://teien.tokyo-park.or.jp/en/kiyosumi/index.html
Yumenoshima Park 夢の島熱帯植物館
Free entry on May 4th.
(Japanese only) http://www.yumenoshima.jp/index.html
The Shiba Detached Palace Garden 旧芝離宮恩賜庭園
Free entry on May 4th.
http://teien.tokyo-park.or.jp/en/kyu-shiba/outline.html
Jindai Botanical Gardens 神代植物公園
Free entry on May 4th.
https://www.tokyo-park.or.jp/english/park/detail_03.html#jindaibotanical
Mukojima Hyakka Flower Garden 向島百科園
Free entry May 4th.
https://www.tokyo-park.or.jp/english/park/detail_07.html#mukojimahyakkaen