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Baby drink
http://www.destinationbw.be/fr/bois-du-beumont in Wavre has BBQ spots... you need to get there early to grab a place (I think)
it includes grills and awnings, etc.
I'm quite familiar with the Lier/Duffel area (at least 60kms from Wavre!) and I don't know anywhere like you describe.
Vrijbroekpark on the western edge of Mechelen may well have what you are looking for and Hofstade - much bigger and south of Mechelen - will certainly suit you though it can get very busy during the school holidays. There is also the Nekker Domein on the eastern edge of Mechelen but I have no idea of the facilities. The Mechelen city website will probably have details.
http://www.visitlier.be/fr/que-voir/nature/parc-municipal
In Lier- states that they have a playground next to a place to picnic...
State where you live.