I've done this before. I took my Vonage box and my CallVantage box with me on two separate trips.
The trick with Vonage tends to be that it wants access on a bunch of ports. So you need to port forward stuff - or stick the Vonage box in front of the router. The symptom of not doing this was that the call would either sound really bad, or initially sound good and then abruptly terminate after a couple of minutes.
So, my answer is that you can do this if you can plug the Vonage box directly into the internet connection of wherever your parents will be. So if it's a house or something, just do modem -> Vonage box -> router. If it absolutely needs to be behind a router (like a hotel's connection) then it may work but you will likely have problems. Vonage uses a lot of devices and maybe other ones work better than the Dlink one that I was using.
One thing that I will say is that Vonage technical support is no help at all if you have issues during international travel with it. Their very prompt and unequivocal answer is "we don't support traveling outside of the US with our telephone adapters." which is a reasonable enough answer, but it's not very helpful when they won't even help to troubleshoot it at all.