No, it is not safe to use a VapoRub product 'near' a parrot.
Danger - Direct Contact: Petroleum Jelly will coat the feathers and result in the parrot working (beaking) the feathers in an attempt to clean them. Ingesting 'petroleum' products is dangerous for humans and very dangerous for parrots. You will note that petroleum jell is not listed as an active ingredient, since it is considered a carrier in this human use; external. Once it is on the beak it is inside the beak seconds later.
Danger - Vapors: The Active Ingredients of a VapoRub is (by volume): Camphor 4.8%, Menthol 2.6%, Eucalypyus Oil 1.2%. This stew, when tightly held against healthy human skin or simply placed on sensitive skin areas (open wounds, inner nostris, etc...) will cause a mild to server irritation (think burn). In air sacks, it is possible to consider server irritation to the exposed inner surfaces.
Proper use of this product is direct application (external use only) to the healthy skin of the chest and throat. Never place this product on the skin directly below or on the nose. This product is not approved for non-human use.
When considering the distance between the (application areas) throat (8") and the chest (14") estimated, adult, and the nose. It would be safe to consider that a parrot should not come any closer then 30" from the application area and than only for 'very' short periods of time, seconds not minutes.
Hands must be washed prior to contact with pets. Whether feathers or fur, all will use their month to clean the contact areas.
I'd exercise good judgement about it. I would keep whomever is using it away from the birds and be sure to wash your hands really well if you apply it to someone else. I don't think someone using it in another room will be harmful.