You might assume that honoring your parents is the kind of natural good deed which should apply equally to Jews and non-Jews, like not committing adultery or establishing a fair court system. But interestingly, there are sources which seem to indicate otherwise. The mitzvah of kivud av ve'em appears in the first half of the Ten Commandments. The normal division of five and five would thus place this command among the mitzvot bein adam lemakom, not those bein adam lechavero. And out of the Ten Commandments, only kivud av ve'em and Shabbat do not find parallels in the seven commandments to non-Jews, which are often identified with natural morality. Why?
Perhaps one important aspect of kivud av ve'em is the preservation the covenant. A main function of the Jewish parent/child relationship would be to transmit religious beliefs and behavior to the next generation, and respecting your parents would facilitate this. Since non-Jews do not need to transmit Judaism to their kids, a lesser degree of parental respect would be required of them.
Certainly one of the major roles of a Jewish parent is to provide the children with religious knowledge and consciousness. In several places (shema, sipur yetziat mitzrayim), we find the parent specifically is commanded to teach Torah to his children. Many Biblical verses testify that this is not an occasional obligation, but rather the parent should be the primary source of religious knowledge. "Ask your father and he will tell you, your elders and they will say to you" (Devarim 32:7). "Hear, my son, the guidance of your father; do not dismiss your mother's Torah" (Mishlei 1:8). God is often referred to as "the God of your fathers" - without revelation to the fathers (parents) and transmission to their offspring we would not have a relationship with God. Religious people (Melachim 2 2:12) and even kings (Melachim 2 6:21) refer to prophets as "father" - this particular term of respect being chosen in reference to the prophet's role as a religious teacher.
Halachically, the verse "Mipnei seivah takum" is interpreted as not just the older generation in general, but specifically Torah scholars, those who transmit the covenant. Many of the laws of honoring your rabbi are similar to those honoring your parents, hinting to a common reason for the respect. And perhaps, the punishment for cursing your parents is death, because it means cursing the source of your religion, and thus is tantamount to blasphemy. In each case, the parent's special position in the commandment comes about in connection to the teaching of Torah and transmission of Judaism.
More speculatively, we might go a step further and suggest that something is fundamentally lacking in parenthood which does not involve the teaching of Judaism. Halachically, the laws of inheritance do not apply to converts to Judaism. Is this because, as usually explained, converts are "born anew" and thus are in some sense not related to their biological families? Or more radically, perhaps the Torah does not fully recognize non-Jewish parenthood at all (since it cannot contribute to transmission of Judaism) and thus non-Jews fundamentally have no inheritance whatsoever? (Admittedly, there are major problems with this hypothesis.)
In any case, even if kivud av ve'em is counted one of the mitzvot bein adam lemakom in the Ten Commandments, it is also the very last one of them, immediately adjoining the mitzvot bein adam lechavero and apparently serving as a bridge between the two groups. Whatever the ritual dimensions of the commandment may be, you may not neglect the natural moral sensibility which should go into your relationship with your parents.