Can you make PRSI (or equivalent) contributions in two countries at the same time?
Just to add to the informative answer provided by delfio; it also depends on which two countries, and specifically the rules from the country in which you wish to opt in to pay the voluntary contributions.
If you come from the UK to Ireland (or any other country), you can opt in to pay UK voluntary contributions (upon meeting the aforementioned 3 year rule) and shouldn't have any problems even if you are paying your compulsory PRSI in Ireland.
But if you went the other way, from Ireland to the UK (or any other country in the EU), then Ireland would likely not accept your voluntary contribution application if you started to work and were paying UK National Insurance.
Of course, the UK is leaving the EU on 29th March 2019, so possibly after this date it would be possible.
I'm interested and curious in reading other people's experiences of paying UK voluntary contributions whilst living in Ireland (or elsewhere), and if they are building towards a future of claiming both the 2 full state pensions.
As for myself, well I'm in my 30's, I've been working in Ireland for just over 10 years, so I've just passed the 520 PRSI contribution mark, which I guess will get me something from Ireland at old age, no matter where in the world I may be. I only have a handful of UK contributions to my name, but I'm on the verge of making a lump sum payment to UK Revenue to purchase a handful of backdated years.