Besides the park per se, there are a number of montane forest locations around Oxapampa that bear mentioning.
To the west are two great ACPs (Private Conservation Areas), both with delightful owners who are birders in their own right.
Bosque de Churumazú, right outside the small town of Chontbamba, is barely 15 minutes from Oxapampa and has a couple of nice trails. Despite its small size, it is connected to a larger tract of forest and in a couple of years it's already amassed a very respectable bird list. The whole valley, sitting around 1800 masl, is notable for having lowland species turn up at higher elevations than usual, for example Swallow Tanager, Ruddy Ground-Dove, Magpie Tanager, or White Hawk. I even had a Crowned Slaty Flycatcher right on the edge of town this summer! Two further example that are regularly seen at Churumazú are Black Hawk-Eagle and Chestnut-eared Araçari. This is also the best spot in the area for Chestnut-backed Antshrike (pic 1).
One headache that you'll likely become acquainted with when in the area is how to distinguish the endemic Green-and-White Hummingbird, from the nearly identical White-bellied Humminbird. Personally, I'm still not convinced that they're distinct species, but Churumazú is a reliable spot for the endemic.
Finally, I haven't seen either of those personnally, but Eduardo, the owner has photographed both Chestnut-crowned Gnateater and Mitred Parakeet, two species that are rarely reported elsewhere in the region.
Farther to the west, and perhaps 45 minutes from town, is El Palmeral, named for the abundant Dictyocaryum lamarckianum palms that grow in the area (it's one tree that is generally not cut down in the region as the local honeybees are very fond of its pollen). The trails are not as extensive and the bird list somewhat shorter but the house is absolutely gorgeous, and Patricia, the owner, rents out a couple rooms to visiting birders. It is also without doubt the best place to look for the endemic Creamy-bellied Antwren (pic 2).
Also, if you head there before dawn, the grassy wetlands and cane plantations a few km before arriving are a good spot to look for Striped Owl. Meanwhile, Blackish Rail is practically guaranteed. The latter is described as rare in Birds of Peru, but it's abundant in the area, even being found in the overgrown grasses of the abandoned airstrip that dissects Oxapampa!
Meanwhile, if you continue on the road past the entrance to El Palmeral, you enter a sector named La Suiza. The road continues for a dozen kilometers, crossing a river (make a right after the river) and going up a ridge to some stunted forest topping out at 2850. There aren't that many birds on the higher reaches, and all of those can be better seen in Santa Barbara (whill I'll cover in a later post).
However, the lower parts are very rich and birds, and several things seem to be easier to see on this side of the valley than they are on the other side, where San Alberto is. These include Golden Grosbeak (pic 3), Fawn-breasted Brilliant and Blackish Antbird. It's also a chance to see four flowerpiercer species side by side, namely Masked, Bluish, Deep-blue, and Rusty Flowerpiercer. At times, Hooded Tinamou can be quite vocal, though of course, good luck with actually seeing it! This may also be the best place in the area to connect with the White-rumped Hawk (I've seen it twice there - pic 4), and it's the only time in my life where I've seen a perched Short-tailed Hawk. It's also also one of only two spots where I've seen the scarce Rufous-capped Thornbill.
If you go south instead of west in Chontabamba, you'll get to the famous Ulcumano Ecolodge. It has been visited by a number of high-flying birders and has acquired quite an impressive bird list over the years, though I have to say that I had rather terrible luck both times I was there. The main draw there (which I haven't seen since I haven't spent a night here) is the Cloud-forest Screech-Owl.
Crossing back to the other side of the valley is the famous Carretera Antena that Josh mentioned. It's in absolutely dreadful condition, and should only be attempted with a sturdy 4x4 and a driver who knows what they're doing, and it will take about 1h30 to get to the ridge where the good stuff is. It seems to be the only place around here to see Mountain Velvetbreast. Along with the usual mountain suspects, keep an eye out for Rufous-headed Pygmy-Tyrant , which is generally scarce around here, as well as Black-capped Tyrannulet. We also once startled a Swallow-tailed Nightjar that was on the road.
A bit farther east, on the old road to Villa Rica, is the Bosque de Sho'llet, a stunted forest on a ridge at 2400 masl. It has a lot of Chusquea, and both Barred Parakeet and Slaty Finch can be abundant if the bamboo is seeding. It's also good for Long-tailed Sylph, Glossy-black Thrush, and Variable Antshrike.
Some typical montane species which I've so far forgotten to mention are Band-tailed Pigeon, the delightful Cinnamon Flycatcher, and Smoke-colored Pewee.
Along with the ridge itself, I can recommend stopping around the km 4 marker on the road to Sho'llet. There's a nice mixed flock that hangs out around there, which has stuff that I've hardly seen elsewhere, Jet Manakin, Blue-browed Tanager, Blue-winged Mountain-Tanager, and Fulvous-breasted Flatbill (pic 5) in particular. There's also a roadcut where a pair of Cliff Flycatcher can often be seen hawking for insects.
Finally, the Laguna San Daniel, an hour to the north of Oxapampa, is another sector of the park that is sometimes advertised to tourists. However, the track to get there is not well marked and the lake is tiny, without any waterbirds. The only thing of note seem to be Yellow-rumped Woodpecker, with most of the records of the species for the region.