The Legislature grounds and the Museum (Government House) are great locations for portraits and/or wedding groups. The nice old architecture, with plenty of steps, makes for a great place to photograph groups. I wouldn't suggest that for a toddler though...especially in the summer. There are hundreds or even thousands of weddings groups there every year...so it's a very busy place...especially on Saturdays. Might not be too bad during the week though.
What type of photos are you looking to take? I don't see those locations really being great for photos of a toddler...I would think a park or playground or something like that would be better. In which case, any location would do...even a front lawn.
Do you want something with a dramatic background that says 'I was in Edmonton'? If so, I would suggest somewhere on the south side of the river, overlooking downtown. There are great views starting at Gallagher Park (top of the hill)...and then moving East down Strathern Drive. In the summer, there are usually some great sunsets to accompany the city skyline.
Other options would be Hawrelak Park, Goldbar Park, Rundle Park (where I had my wedding) etc.
Looking at a map, there are many, many parks...especially through the river valley...I'm sure you won't have trouble finding something.