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1. Dear Evghenia, yes, you are correct about the law in this area. Should you obtain UK citizenship, you would be entitled to bring your Moldovan parents to live with you should you move to live in another EU country. For example, should you, as a UK citizen move to Ireland, you would be able to bring your Moldovan parents to live with you as a dependent in Ireland should you so wish.
2. This is allowed under the EU Free Movement of Persons principles. However, be aware that this rule applies uniformly in each of the Member States of the EU. So, if you were to move to France, or Germany, for example, you could still bring your Moldovan parents to live with you in each country as well, so long as they weren't a burden on the state ie were financially able to look after themselves or you were able to look after them.
3. Be aware that there is no EU country which would provide for benefits to be paid to your parents in this situation. Instead, they would have to become naturalised citizens of that country (after between 5-10 years) and then they could access the benefits system and health system of the country in which they were then citizens of. So, the answer is that ther e is no country which is more generous to your parents if they were to move there. The only difference would be that some countries, such as Germany, would not allow them to become naturalised citizens, whilst other countries, such as Ireland, would allow them to become naturalised citizens after five years residence.
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5. Your parents would be able to join you immediately you move. There is no need to wait several years. Once you are resident in the new country they can join you. Secondly, there is no "proving" of ability to support your parents. You merely need to apply for them and they must be able to support themselves. This means that they will not get benefits. Thirdly, YOu will need to consult each States law to find out about the individual rules of naturalisation. Here you can also check the EU websites.
6. What I state "here" I mean "on this subject". Essentially, you can search the websites of the individual counties to see what are the laws in the State in regard to gaining citizenship when you are an elderly person residing in the country.