Surg,
I see you're as controversial as ever
Birmingham as some good parks and open spaces. Their parks department does a splendid job each year in bringing colour to the city. The main routes into the city are decorated well with seasonally blooms and spring bulbs on most road islands.
I'm not sure whether Sutton Park would fall under the remit of 'within the city boundaries' but if you get the chance it's worth considering a slight detour before you get back onto the M42 to head back north.
Parks worth considering are Pype Hays, Handsworth, The Botanical Gardens, Cannon Hill.
The parks department are frequently commended most years at Chelsea Flower Show if you were to contact them I'm sure they will be most helpful in directing you to the best areas depending on the time of year of your visit.
Cannon Hill was the venue for many years of the garden programme
whose name escapes me.
At present garden programmes feature the area of Bourneville, one of the first garden suburbs.
Birmingham has very well subscribed allotment sections dotted about the city with large waiting lists for plots. The largest diverse one is in Handsworth and has been featured often on garden programmes. You can find many secluded areas by viewing the city with Google satellite.
A little further a field worth considering is Walsall Arboretum they have a flood light evening walk in the Autumn which shows off their collection of trees very well.
I hope you find some gems and share them here.
An amend.
If you come into the city via Spaghetti Junction you'll drive through the largest collection of Lupins in the world