What about PNR agreements with third countries? 

Agreements for the transfer of PNR data have been concluded between the EU and the USA, Canada and Australia. Another PNR deal is currently being negotiated with Mexico. The European Parliament gives its "consent" to these agreements (it can approve or reject them as a whole, but not amend them).

The EU PNR proposal is for a directive and the co-decision procedure applies, meaning that MEPs can table amendments to the Commission proposal and negotiate the text with the Council and the Commission (in "trilogues") before the text becomes law.

EU-US: the agreement entered into force on 1 July 2012, replacing a previous one dating from 2007. The European Parliament gave its consent in April 2012. Press release

EU-Australia: The European Parliament gave its consent in October 2011. Press release

EU-Canada: Parliament referred the agreement to the EU Court of Justice (ECJ) in November 2014 for an opinion on whether it is in line with the EU Treaties and Charter of Fundamental Rights. Parliament’s final vote of consent will be adjourned until the ECJ has delivered its opinion. A previous EU-Canada PNR agreement from 2006 remains in force until a new one can replace it. Thus the delay caused by seeking the Court's opinion does not result in any security gap. Press release

On 14 July 2015, negotiations for an EU-Mexico PNR data transfer deal were formally launched (the Council has given a mandate to the Commission on 23 June 2015). The plans have been debated in plenary on 15 April 2015 and in the Civil Liberties Committee on 4 June 2015. Any draft PNR agreement with Mexico will need to take into account the content of the ECJ opinion on the EU-Canada PNR deal. Once concluded, it could enter into force only with the European Parliament’s consent.