Welcome! I'm in Broken Arrow too! I have a 3 1/2 year old and an 18 month old. I like Ninehius Park also they have a great wading pool with preschool size slides an water things in them. Central park (on Main between 81st and 71st) has a nice wading pool area also and a great playground.
I like Heaven Fun for an inflatable jump place. It is at 71st and Lewis in Tulsa, it is about $7.00 (I think under 2 is free) to jump as long as you want. There is a place in Broken Arrow called Pump it Up, but I haven't been there yet.
Other things we like to do are go the the Tulsa Zoo($6 adults, under 3 free), Oklahoma Aquarium in Jenks ($20 for Adults, 3 and under free), Tulsa Air and Space Museum ($6 adults, 3 and under free) (it is kind of small, but the kids can get in an old airplane and my son loves it), Safari's Wldlife Sanctuary (in Broken Arrow go E on 71st to Oak Grove Road and then follow the signs, They have alot of big cats(lions, tigers etc) you can feed the cats chicken from a pole or feed the other animals animal crackers Adults $6, 3 and under free).
Outdoors things we like to do is hiking at Ray Harral Nature Park (121st St. between Elm and Lynn Lane), hiking at Oxley Nature Center (in Tulsa at Mowhawk Park near the Zoo), hiking at Claremore Conservation Reserve at Rogers University in Claremore OK. Berry Picking at Toomeys Berry farm (61st off of Hwy 169) Thunderbird Farms at 301 E. Ave and 71st St).
These are just a few things and there are many more. I would also check out Tulsa Kids magazine for seasonal events(www.tulsakids.com).
You might want to join a Mom's Club. We are members of Tulsa Family Club ($20/year) www.tulsafamilyclub.org. I also go to a MOPS(Mothers of Preschoolers) at First Baptist Church of Broken Arrow(61st and Elm) which is like a Mom break where screened teachers watch your preschoolers while mom's have a potluck breakfast and talk with other moms about families, and make a fun craft item. It is $5 each meeting(covers cost of the craft) (Sept-May 1st and 3rd Tuesday of every month). It is great fun and I have made alot of friends there and had some great advice from other moms)(www.mops.org).
I hope I didn't overwhelm you with info.
KT