What equipment does an A320 need in order to operate under RNP 2?
According to ICAO/EASA till 12/2015 Version.
And from Airbus getting to grips with modern navigation.


  Regulatory Authorities have published standards for specific
  airspaces. Provide certification and operational requirements for
  Basic RNAV (RNP-5 Type) and for Precision RNAV (RNP1 Type) within
  European airspace (refer to Annexes 5.1 and 5.2 at the end). FAA Order
  8400.12A addresses RNP-10 operations in oceanic and remote areas (refer to Annex 5.3 at the end). Also RNP AR required equipment is
  specified in several sources and the FCOM manual of airbus. In
  addition for RNP 2 specific terminal and approach operations the
  requirements are specified on the approach charts of airports that implementing it.


Most of the operators/airlines are using the standard required equipment for RNP 1.


  According to FCOM A319/A320/A321 FLEET / PRO-SPO-51 P 8/10 The minimum navigation equipment required to enter RNP 1 airspace is:
  
  ‐ One FMGC
  
  ‐ One MCDU
  
  ‐ One GPS
  
  ‐ Two IRS
  
  ‐ One FD in NAV mode
  
  > ‐ Two NDs (the temporary display of ND information via the PFD/ND
  switch is permitted on one side).


Some helpful information also about RNP 2 you can find also here: http://www2010.icao.int/NACC/documents/meetings/2013/pbn2013/ops7-en.pdf

Moreover Airbus company under PRO-SPO-51 P 5/10 in FCOM refers as RNAV1(2)
so it takes exactly the same equipment for RNP 1.

Let me know if something is different, or if you have different information from another specific-related documentation.