There is a third link: Rabbi Yehuda ben Dama, one of the ten martyrs. He seems not to be mentioned anywhere else.
This is complete and utter speculation, but here is how I interpret the "poetry" of the lesson:
Eisav's (and therefore Rome's) big merit was honoring his father, but only superficially as he planned on killing Jacob after Isaac died (Genesis 27:41). He only honored his father such that he could receive favoritism.
Dama ben Netina's honoring of his parents was greater because Dama's parents were both insane. He had no chance of receiving favoritism from them. In fact, he took a financial hit.
The entire Roman Empire observed his exemplary honoring when his mother ripped off his clothes while he was addressing the Senate. The senators surely laughed at the spectacle and mocked Dama, degrading the commandment of honoring one's parents and repudiating their merit.
A generation later, Rome does not try to increase their commitment to their ancestor's mitzva and executes Dama's son. Dama's inclusion in the Tisha b'av readings is a comfort because when they executed him, they guaranteed their eventual downfall. Hadrian became Emperor at Rome's greatest extent (117). He was emperor when the Bar Kochba revolt took place in 135. The empire would decline constantly until the end.