My fiancé insists that after marriage I (New Muslim) should snap relationship with my parents even though they are fine of my marrying into a Muslim family. Is this correct as per Islam.?

The boy's family has put up a wrong condition. The stand of the boy's family is wrong. One shouldn't keep relation with one's family only if they prevent one from following Islam. If they do not put obstacles in your following Islam, then you can maintain normal relations with them. This principle is derived from the following verse of the Quran: "We have enjoined man to show kindness to his parents. But if they bid you associate with me something about which you have no knowledge, do not obey them." (29:8).

Therefore, if your family has no problem with your Islam, even then the boy's family puts the condition of not maintaining relations with your family, then the stand of boy's family is incorrect.