There are few mitzvoth more important than that of kivud av veim, honouring parents. “Our Rabbis taught: There are three partners in [the creation] of man. The Holy One blessed be He, one’s father and mother. When one honours their father and mother the Holy One blessed be He says: 'It is as if I dwell amongst you and you honour Me” (Kiddushin 30b) Is it any wonder that the mitzva to honour one’s parents is placed in the aseret hadibrot alongside the mitzvoth between man and G-d? Nor should we be surprised that it is one of the very few mitzvoth where we are promised a reward of long life.
While an atheist can honour his parents, one who fails to do so demonstrates by his actions, or lack thereof, that he does not truly believe in G-d. “When one brings pain to one’s parents The Holy One blessed be He says: ‘It is good that which I did that I do not dwell amongst them for had I dwelled amongst them they would have caused pain to Me.” (Kiddushin 31a) 
Ironically our Sages assert that it is specifically because G-d shared the honour due Him with flesh and blood that the nations of the world gained an appreciation of G-d. “At the time when the Holy One blessed be He declared ‘I [am the Lord your G-d]; there shall be [no other gods] the nations of the world said: ‘His own honour He seeks’. When He said ‘Honour your father and mother’ they accepted [the truth] of the first teachings.” (Kiddushin 31a) The rabbinic descriptions of G-d are meant to teach those created in the image of G-d how to act. “Who is honoured? Those who honour all living things.” (Pirkei Avot 4:1)
With honouring our parents being of universal value our Sages choose examples of non-Jews to teach us the importance of this mitzva. This is striking and tells us much about the world-view of the rabbis. The Biblical paradigm for this mitzva is none other than Eisav. Whatever faults he may have had our Sages did not hesitate to extol the great honour he gave his father, Yitzchak. The same Mishna that teaches the importance of honouring all who have been created begins by asking “Who is wise? He who learns from everyone.” Everyone encompasses a wide net, including idol-worshippers. They may be guilty of the most serious infractions against G-d and one that so much of the Torah is devoted to the eradication of, but that does not mean we can’t learn from them. They may actually do certain things better than we.
“It was asked of Rav Ulla: How far [must one go] in honoring parents? He said to them,’ Go out and see what one idol worshipper did in Ashkelon and Dama ben Netinah is his name.” It is fair to presume that this was not the answer the questioners were expecting. But sometimes it's best “to go out” from our comfort zone and “see” what we might learn from ‘outside our little bubble.’  
The Talmud describes how Dama ben Netinah turned down a lucrative business deal as it would have entailed waking up his sleeping father[1][2]. “In a future year the Holy One blessed be He gave him his reward and a red Heifer was born in his herd.” Finding a red heifer is next to impossible. Yet without it once someone came in contact with a dead body, say by attending a funeral, they would be unable to enter the Temple. It is no exaggeration to say the entire Temple ritual rested upon having a red-heifer. 
The combination of little to no supply and high demand allowed Dama ben Netinah to say to the Rabbis “I know that if I were to ask for all the money in the world you would give it to me.” However he did not want to take advantage of the situation and continued “I only ask for the money I lost due to the respect I showed to my father.” There is much to learn from some idolaters[3]. The Talmud then notes the comment of Rabbi Chaninah “if one who is not commanded acts such, one who is commanded how much more so.” [4]
The reward of a pariah adumah seems to be a rather strange one. The simple explanation is the one alluded to above. While meaningless to Dama ben Netinah [5] it was most valuable to the rabbis allowing him to recoup his foregone income - and then some had he chosen to do so. Yet it seems the connection runs deeper.
Kivud av veim and parah adumah are polar opposite mitzvoth. It is hard to imagine a more readily understood mitzva than honouring those who brought us into the world. And even if they neglected us we are still indebted to them. On the other hand there is no mitzva more difficult to comprehend than that of the parah adumah. Does it make any sense to find a red cow, kill it, burn it and sprinkle some of ashes upon those who came in contact with a dead body, specifically on the third and seventh day? Almost as strange, this act of purification makes the one who carried it out impure. 
Yet G-d fearing Jews observe both mishpatim and chukim, the rational the seemingly irrational. It is the same G-d who commanded them both. Sprinkling ashes may not offer as much meaning or even be as important as honouring parents. But both emanate from the same divine source. 
And truth be told what is rational is not always clear. Does it really make sense to let your father sleep and forego in the words of the Talmud “800,000 golden dinarim”? While such is undoubtedly an exaggeration what father would not want to be woken up so his son could make lots of money. 
And at times the mitzva of honouring parents is beyond rational. Our tradition demands we honour our parents even when they are not deserving of such. And again is Dama ben Netinah the idolater from Ashkelon who is our role model.
“When Rav Dimi came, he said: He [Dama son of Netinah] was once wearing a gold embroidered silken cloak and sitting among Roman nobles, when his mother came, tore it off from him, struck him on the head, and spat in his face, yet he did not shame her.” 
Rational or not we must display unconditional honour to our parents.
[1]The laws of honouring parents likely do not require one to forego such a gain in fulfillment of this mitzva. Dama was not interested in just following the law but in going beyond it. The sleep of his father was worth more than all the money in the world.
[2] While not the focus of the story one can also learn how important it can be to act quickly when opportunity presents itself. How long can one sleep? Yet by the time the father awoke the deal was no longer viable. 
[3] What makes this all the more fascinating is that according to many authorities with such an inelastic demand one is allowed to charge whatever the market will bear. For a discussion on pricing policies in such cases - i.e. water in the desert, armaments in the battlefield - see Levine, Aaron, Case Studies in Jewish Business Ethics, chapter 4 pp. 153-163.
[4] While this comment would seem to be reflecting on the great kivud av shown by this idolater the comment appears immediately after Dama refused to charge an exorbitant price for the red heifer. I would like to think Rabbi Chaninah is reflecting on his business practices also. It is to be expected that the business practices of the Jew be of highest standards even if a lower standard is well within one's legal rights which as noted above is a matter of dispute - as far as Jews are concerned. No authority demands a non-Jew legally refrain from charging what the market will bear.   
[5] His name is not coincidental. Dama is related to word damim, meaning money, and Netinah means to give. Dama ben Netinah was willing to give lots of money to honour his father.