Applying for PhD programs abroad while doing one in your home country
It's usually a bad idea to do two PhDs at the same time because:


A PhD is a full-time job, with some very intensive periods. One has to deal with enough stress already when doing one PhD.
It would unavoidably generate conflicts: sometimes your advisor no 1 will ask you to do something which will keep you busy the full week or month. What do you tell your advisor number 2, that you're on holiday? If your two advisors don't agree on the direction of your research, what do you do? What if one insists to have their name on a paper you did with the other?
Either you do your best on both, but then it will take you more time than doing only one, probably double. Or you do one seriously and you botch the other, but then what's the point of doing it?
Assuming you survive the two PhDs, at the end you can apply for a postdoc or industry job... Exactly the same as with one PhD, no benefit.