M.N.
Hi H.!
My daughter is almost 2, and around 9 months old, I found several that were perfect for toddlers! I'm not sure where you live, but there are several that are central and are lots of fun.
My favorite is Little Stacey Park (next to Little Stacey Pool) in Travis Heights neighborhood off Congress (near Riverside). It has great shade, lots to do all geared to under 5, and a small pool that's lifeguarded and is a wading pool for toddlers open during the summer.
I live in French Place in East Austin, and I also love Patterson Park which has great covered table areas for parties and the playground is so nice and perfect for 5 and under too.
There's also a small park that I recently discovered but, would be perfect for toddlers and a party and not crowded. It's got the fun fountain area where fountains squirt up at different levels and toddlers can run through them and a great playscape and a fun covered area with picnic tables. It's small but, perfect for a toddler's birthday party. If you take Manor off I-35 and go East for about 1/4 mile, take a right on Cherrywood, it is about a 1/4 mile on the right. My daughter loved playing there and we stayed about 2 hours. It might also be the perfect birthday party park for privacy - not too many people seem to know about it.
Here's a link of Austin parks - if you look through the maps, each one has photos of the individual parks too!
Hope this helps.
M.
http://www.ci.austin.tx.us/parks/parkdirectory.htm