Alpha Penking is inside the Sealed Realm of the Frozen Wings, entered around (114, -352). The nearby fast-travel statue is around (119, -337). Travel to the statue, then head south to the entrance; the statue itself does not start the fight.
Find the dome, not a roaming Penking
Look for the dome's interaction prompt near the entrance coordinate. Penking is inside, so searching the surrounding hillside for a roaming boss will take you away from the right place.
The dome is also marked around (114, -351). Once you are close, use its interaction prompt to finish the approach.
Read the prompt before committing supplies. If the encounter is unavailable, follow the message in your own world rather than a respawn countdown from another server.
Make room for the Pengullet
The encounter pairs a level-15 Penking with accompanying Pengullet. Penking is Water/Ice.
Look at the group before focusing on the Alpha. If smaller attackers keep interrupting your aim, deal with that pressure first. Keep the enemies in view and move into open floor space instead of backing into a wall while lining up a shot.
In co-op, decide who will handle the smaller enemies and who will call the capture attempt. It is much easier to control Penking's final bit of health when everyone knows when to stop attacking.
Catch Penking or farm its rewards
For a capture, bring a weaker attack for the end of the fight. Once the smaller enemies are under control, weaken Penking, recall any Pal that might finish it, and check your sphere's displayed capture chance. The damage that made the opening comfortable can be too much for the final throws.
If you are farming the Alpha, its reward table lists Feathered Hair Band Schematic 4 at 3%. Expect repeat attempts: the listed chance does not increase because the previous run missed.
If you just want a Penking for your collection, check breeding options when you already have suitable parents. This fixed encounter is one way to get the species, not its complete habitat.
Browse the Pal location finder for another target. Chillet is an outdoor alternative with a more open approach.