Booth Park was newly renovated in the heart of Downtown Birmingham. Bring a piece of cardboard because there is a turf hill that allows for kids surf down. Everyone is doing it! (Not only one infant swing, one special needs swing on site). Many special events occur at the park during the summer months. Check the Birmingham City Page for details.
It is a wonderful and clean park. It is in walking distance of everything. Bring your stroller for a walk uptown for some coffee, and bring it to the park.
The only draw back is that if you have multiple young child, there are obstructions from watching your child. And it is not fully fenced in, and it is on Old Woodward (a few hundred feet away).