I was in BRU last week (16-17th) and while the airport was very busy there was ample parking space in the T1 Frontpark.
So while the spaces that you can reserve are booked out, I'm pretty sure you can find a space somewhere.
If there's another adult in your party, I would first drop off all passengers + luggage at the departures, then leave the airport, turn around at the roundabout (petrol station) and go back in, park at T2 or T3. If both are full, park at T1, only slightly more expensive.
You can also just park immediately, there's trolleys for luggage in the parking lots.
http://www.interparking.be/brusselsairport/Compare.aspx