I'm a wine professional, living in Bangkok and travelling frequently throughout South East Asia. I'm a huge fan of Laos and Cambodia and off the beaten track travel. Feel free to contact me at: www.facebook.com/mil.elephants
Bangkok is lacking open and green spaces, thats for sure.
The few parks that can be found in the centre of the city are often very busy and not necessarily relaxing. One exception is Suan Rot Fai or Rail Way Park not far from Jatujak Market and Park. It's formally known as Wachirabenchathat Park, although I know no-one who calls it this.
It's just about the only place I know in the city that has a feel of being in the country, and is easily reached by BTS (Mo Chit Station) or MRT (Kampeang Phet Station). Sure, you can still hear the buzz of the city in the distance, but you can also rent bikes and rowing boats, feed the ducks and enjoy the green spaces.
It's my favourite space for a nice romantic walk in the city, if that's your inclination, or to take the kids, kick a ball or eat some ice-cream. There are some food stalls nearby, but I would recommend taking a picnic and making a day of it. It's busier on weekends due to it's proximity to Jatujak Market, but there's still lots of space.
Right next door to the park are Queen Sirikit's Gardens, which are more preened and feature a lot of beautiful flower beds and interesting garden designs, but the more rustic feel of Rot Fai appeals to me.
It's great for a nice long walk, but beware most evenings it can be very busy with cyclists racing each other around the main looping path in the centre of the park. There's also more wildlife than you'll see in most parks in Bangkok, lots of birds, large and impressive lizards and quite a few squirrels.
The park can be reached in about 20 minutes by foot from Mo Chit BTS or Kampeang Phet MRT stations.