Islam is beyond and above cultures and traditions. In Islam the criteria is Islamic and not cultural. The important thing is to make sure that this prospective husband or wife is a good Muslim. Race and ethnicity are not considered. However, we must recognize that some difficulties may arise due to some differences in customs, etc. These must be known and addressed in the beginning.
If the parents are not accepting we must know what are their reasons? Are the reasons Islamic or just pure taste? It is therefore, important for the son/daughter to establish communication with his/her parents. If he/she is not able to communicate with them on his/her own, he/she may be better advised to seek other channels in the community to talk to them. Anyway one is required to deal most affectionately with his/her parents, and seek to please them in all other ways.
elaborating on the issue of inter-cultural marriages in general, we’d like to cite the following fatwa:
“Prophet Muhammad (peace and blessings be upon him) has told us that “there is no difference (superiority) of an Arab Muslim over a non-Arab Muslim or of a non-Arab Muslim over an Arab Muslim except in the level of taqwa (or God consciousness and fear).” Prophet Muhammad (peace and blessings be upon him) also instructed the Muslims to choose their spouses based on the level of piousness and not based on wealth, family lineage, or looks. Also, cross-cultural marriages do have their built-in problems since you would be sharing your life with a person (and his family) who come from a totally different culture, so, try to be clear about the issues that may cause problems in the future.”
excerpted, with modifications, from:
Focusing on the authority of parents to oppose a marriage of their daughter due to cultural differences, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states: “Compatibility entails a person’s worth in a spiritual and moral sense: the only primary criterion that makes or breaks a marriage. The Prophet (peace and blessings be upon him) said: “If a person of acceptable religion and character presents himself for marriage, marry him, otherwise, there would be widespread sedition and rampant corruption in the land.”
therefore, if a woman has made a choice based on the above consideration, then her parents have no right to stop her marriage. Since it is her choice to eat the food she wishes to eat, likewise, it is her choice to decide whom she wishes to choose as a life-long partner. They cannot stop her from marrying the person she wants simply because the person is not sharing her culture or ethnic background.
Parents, however, have the authority to intervene should their daughter choose someone of questionable moral and religious character. Should she proceeds in such a case against their wishes, her marriage is deemed null and void according to the rules of Islamic jurisprudence.
If, however, her parents objection to her marriage is based purely on racial, cultural or ethnic grounds, she is allowed to seek other channels of authority to intervene in such a case, as long as the person of her choice is of acceptable religion and character. The concerned authority is supposed to get the parental consent to the marriage.”
Shedding more light on the issue of cultural background as an element to ensure marital stability, the late Sheikh Sayyed Ad-Darsh, former Chairman of the UK Shari`ah Council, states:

“Marriage is not for fun or experience. It is a life-long relationship. For that reason, any factor detrimental to the relationship should be avoided as much as is possible.

Highly educated males and females should seek partners with a similar educational background. Cultural and family background is very important. Common language is an essential way of communicating. Such things help the two partners to understand, communicate and relate to one another and are factors of stability and success – as are financial independence and the ability to provide a decent acceptable level of maintenance. Again, this is a way of ensuring that outside influences do not spoil an otherwise happy life. All ways and means should be considered, giving a solid basis for a new human experience which is expected to provide a framework for a happy, successful and amicable life.

The questions of common language, cultural background, education and age, etc., are meant, in an ordinary stable context, to maximize the chances of success and stability in a very important Islamic institution, that of marriage. However, considering the particular position of Muslim communities living in minority situations, young Muslims, male and female, are exposed to all sorts of challenges – be they cultural, linguistic, racial or social.

The most fundamental question when choosing a partner is a religious one. As far as language, background, or social position are concerned, these are not significant factors that absolutely must be fulfilled before a marriage can take place. If the prospective partner is of good character, with a strong religious inclination, and the two young people are happy and feel compatible with one another, other considerations are not of such importance.”

To conclude, though sharing the same cultural background is an important element in ensuring the stability of the long-life partnership and avoiding the problems of cultural differences, the most important criterion is the religious one.