Your are correct regarding the overall guidelines as being standardized across all EU Country's. The outbreaks of BirdFlu has resulted in several member states enacting additional requirements, Ireland being one of them. Point is that stricter requirements of assure the health of 'Birds' traveling between EU Country's will likely become stricter due to BirdFlu 'Hot Spots' existing over-Winter (2016/2017 & 2017/2018).
Everything really needs to start 'now' with your Avian Vet in developing EU (CITES) documentation base for your MAC. And, yes the PETS (pet passport) from the US Fish and Wildlife Service 'May Be' part of that document set.
Identification is Step Number One: Whether a band and/or a Micro Chip will clearly need to happen 'at' your CAV! Remember that we are chasing 'Proof' and the signature of your CAV sets that 'Proof' in place. On date, such and such, I, Name Here, as a CAV, number such and such did, etc, etc... I would recommend both, since an open band is the only type that can be place on an Adult MAC.
Most CAV Clinics are switching to photo ID's of their client's Parrots. This is added to the historical medical file of your MAC and with the ID number of the band and/or Micro Chip, that provides ownership history. Having an extended history with your CAV is clearly very helpful. Any prior ownership history you have is always helpful.
Health Certificate: This one can become a moving target if there is a out-breaks of BirdFlu in Europe or North America this Spring and/or Fall 2018 and any over Wintering 2018/2019. At present, the Certificate, once issued has a 10 day useful life and you will be required to register with an EU Avian Specialist.
An increase in active Hot Spots in Both North America and Europe may fully halt any transportation of 'Birds!' At present, there are no wide spread Bands.
In addition, Ireland is currently requiring that Parrot(s) be quarantined in your Home for 6 months prior to departure with your CAV preforming 'onsite' visits. I am not sure if this has already become or will become standard for all EU member states.