After the Bali Bombings, some Australians build a system for quickly identifying remains against DNA from close relatives. The point of it was to ensure that the right remains were being returned to the right family members for appropriate funerary rights. I don't know if it was the first time that this approach had been used, but it was certainly a big deal in Australia and it was the first time that such a thing had been done to identify remains on a mass scale for Australian families.
My understanding of this is that 'not store bought dirt's' allegedly naive comment is not really so naive. Most paternity testing is looking for genetic similarity because the problem of which person would be the child and which one would be the parent is apparently solved already. So the idea is that if there is a 50% match, then the father is the father. In a case where time travel is involved, the mother would still have a 50% similarity, and therefore if you think the problem is solved, then the mother could easily pass for the child.
Of course, if you have a father ALSO with 50% simliarity trying to pull the same trick, a careful check may reveal that something's up, but it's certainly possible that two children (to two different fathers) might have a genetic makeup that matches the parents in this fashion.
As for them being siblings; that's a little more complicated as I understand it and the people who do this victim identification through DNA will have a more complete answer than I would but it's possible for siblings to have much more or much less than 50% of their genetics in common, depending on what parts of the genetic code have been shared by which parent and how close they were to begin with genetically. Siblings with close to 50% similarity would actually be reasonably rare, and one would expect that if there is a reasonably close similarity between the parents in the first place that you'd be looking at around 70% or more similarity (those who know better, post comments and I'll edit accordingly if necessary) in most cases.
So there are limits to how this would work, but in theory it's possible to establish a biological relationship that could be misinterpreted due to relative ages of the parties, but as to whether or not that would establish enough of a legal case for citizenship of a specific country, that is a matter for each country.
England (for example) offers automatic citizenship for the children of expatriates, even if they never set foot in the country (the Australian Federal Government has been running afoul of this problem for the last year or so in dual-citizengate) but other countries have a broad spectrum of laws around this and don't always offer citizenship to families.
I recommend consulting an immigration consultant (yes, that's a thing apparently) before starting your work on a time travel machine as individual results may vary.