I've been to Antwerp within the last couple years and Beauval a little while ago, but not to Planckendael. This category feels very strongly represented at both participants.
The Equatorial Dome and South American Aviary both opened after my Beauval visit, so it is likely that the zoo has become even stronger in this category than I recall it being. I won't detail any of those as I have no first-hand experience of them.
The zoo has no less than five tropical greenhouses, although not all of their inhabitants are likely to count here. The Birds Greenhouse was a particular strong suit, with some very large and densely-planted aviaries, all wire-fronted, amidst a large and immersive walkthrough space. Rarities here include Andean Cock-of-the-rock and Pesquet's Parrot. There are also free-flying birds in the Gorillas Greenhouse, which I found particularly excellent, with several pathways off into the bushes allowing close-up views of the birds, and some waterways in the floor inhabited by stingrays proving a particular highlight. If memory serves, there were also some stingrays, along with such mammals as chevrotains and sloths, in the bird house. The climax of the Gorillas Greenhouse is a former manatee pool now home to large Amazonian fish and crocodilians, for whom it is exceptionally deep. The gorilla enclosure itself is one of the larger that I have seen and decently well-landscaped. Elsewhere in the zoo is a similarly-sized second enclosure with a bachelor group.
The Apes Greenhouse has some nice chimpanzee and orangutan enclosures, with good space and solid climbing (although for the orangs you would maybe want a little more, and the landscaping is a little unimaginative). Also in this building are some mediocre callitrichid enclosures and a Vivarium, which I thought was good, but the inhabitants have likely changed drastically with several of them moving to the Dome. The worst part of Beauval in this category is probably the Sumatran Tiger enclosure, which isn't pictured in the gallery, but struck me as far too small and a little bare. Nearby is a rather good Jaguar enclosure - certainly the best of Beauval's central belt of feline enclosures, and fortunately for them here, the only other one that counts in this category. The zoo keeps multiple species of lemurs in beautiful islands allowing them to climb to remarkable heights on mature trees, and nearby is one of the larger and more attractive Pampas enclosures that I have personally seen (the only place I've ever seen baby Capybaras, and they were adorable). Unfortunately, they do share with some pinioned ducks.
Moving to the southern portion of the zoo, I really like the big Clouded Leopard and Javan Langur cages, spacious with a lot of climbing, and attractively fading into the hillside behind. Tasmanian Devils are always a delight, and their enclosure is quite large and steep. Not sure how many inhabitants of the Australia Greenhouse count, but I am sure the tree kangaroos, who have a very large enclosure, will. There is a basic, but big and pretty, Okapi enclosure, shared with one of Beauval's best rarities: the Yellow-backed Duiker. While the majority of Asian Plains won't count, I suspect the Malayan Tapir and Reeve's Muntjac enclosure, which is really good aside from the mix with pinioned cranes. I think we concluded in the previous thread that the Giant Pandas count here, which would prove an enormous advantage to Beauval as well.
@Maxime - family gorilla enclosure, shared with Patas Monkeys.
@Antoine - new stingray, caiman and other Amazonian fish pool, formerly for manatees.
@SivatheriumGuy - orangutan and gibbon enclosure.
@Maguari - a small portion of the Vivarium.
@lintworm - Jaguar enclosure.
@Maguari - overview of the guereza and lemur islands.
@Antoine - Pampas enclosure.
@Antoine - Clouded Leopard enclosure.
@twilighter - Okapi and Yellow-backed Duiker enclosure.
@twilighter - Tasmanian Devil enclosure.
@TheoV - tapir, muntjac and White-naped Crane enclosure.
@twilighter - one of two excellent Giant Panda enclosures.
Now that I think about it, this may well be Beauval's strongest category. There is so much of excellence, and other than the tigers, nothing of any real weakness. All the above and that isn't even including the Dome, which is one of the 'big five' of European tropical houses!
Antwerp doesn't really have much to answer with. I'm not sure whether or not the Indian Elephants count, but if they do then that is a serious dent here as the enclosure is awful. They of course have some excellent ape enclosures of their own, although slightly weaker than Beauval's, I would argue, with Amahoro the last Eastern Gorilla in captivity being a highlight. There are several relevant offerings in the excellent Reptile House as well, but in general, Antwerp alone has nothing even remotely comparable to Beauval. I will have to wait to see from others what Planckendael has, but from what I have seen in the gallery it doesn't quite seem like enough to turn the tides. 3-2 Beauval for now.