My impression is that parks in Tokyo suburbs are just modern, gardens are rare, and small temples and shrines are everywhere. If you have not finished famous ones in central Tokyo, just find the place near or inside the JR Yamanote circular line. If you avoid giant stations, anywhere near your favorite spots should be OK.
If I dare to pick up, Sendagaya - Shinjuku Gyoen, Komagome - Rikugien, Kyu Koga Garden, Oji - Asukayama Park, Yushima - Ueno Park, Kyo Iwasakitei Garden, Hiroo - Arisugawa-no-miya Memorial Park, Yoyogihachiman - Yoyogi Park, Kita sando - Meiji Shrine, and Kiyosumi Shirakawa - Kiyosumi Garden.
Basically, old Tokyo is around Ginza and Asakusa, and newer Tokyo is around Shinjuku and Shibuya. The south is busy but sophisticated, and the north is opposite. Yanesen -- Yanaka, Nezu, and Sendagi -- would be one of the quietest areas inside the JR circular line. Then, going outward, there spread bed towns of commuters.
In the Tokyo suburbs, one outstanding place should be Kamakura, 0:54 from JR Shibuya by Google Maps. Neighboring spot Enoshima has a small shrine. I think Enoshima and Kamakura is a little set of Miyajima and Kyoto. Both has a good connection between JR Ofuna, 0:41 from Shibuya.
Takaosan is 1:03 from Shibuya. On the way, there is Takahatafudo. It is famous for its fudoson temple, 0:47 from Shibuya. Fuchu is 0:26 from Shibuya. It has a shrine just near the station.
Koganei Park is 0:57 from Shibuya. JR Musashi Koganei is at the south of the park, 0:32 from Shibuya. There is a small garden nearby, called Sorosen-en. Tonogaya Garden is at one station west, JR Kokubunji, 0:32 from Shibuya. Well,.., train times fluctuate as I searched. I mean they are not so accurate. Here I nominated two gardens, but take care of mosquitos in summer.
Inokashira Park station is 0:29 from Shibuya, but it takes only 16 min to Kichijoji by a use of a direct rapid train. Some small temples are scattered over the north of Kichijoji.
Kasai beach-side Park is 0:41 from Shibuya.
Toneri Park is 0:52 from Shinjuku. Nishiarai Dashi Temple is nearby, but it might take a bit more time, as they are at the fully opposite side of Shibuya.
Heirinji temple in Niiza is one hour away from Shibuya.
It takes a similar time to get to Mitsuike Park, but a nearby Ookurayama station is 0:36 from Shibuya.
Well,.., if I take a balance between place values and times from Shibuya, I would list again Yanesen, Kamakura, and Kichijoji, as the top three for a moment.