Darry is ashamed when he sees Paul Holden, because he is a reminder of what Darry might have had.
Darry is surprised to see Paul Holden. He used to be on the football team with Darry. He seems to look down on Darry now, making Darry feel embarrassed to be a greaser.
[You] could see he hated Paul now. It wasn't only jealousy— Darry had a right to be jealous; he was ashamed to be on our side, ashamed to be seen with the Brumly boys, Shepard's gang, maybe even us. (ch 9, p. 143)
Darry does not like the expression of contempt, hatred, or pity on Paul’s face. Ponyboy says that only he and Soda can tell what is going on inside Darry’s head as he looks at Paul, and remembers the life he used to have or might have had, if his parents had not died and left him the head of the family for his two brothers.
When Paul Holden showed up in the rumble it made Darry feel a sort of feeling of jealousness. I could imagine what Darry is feeling at that time, seeing someone who was less superior then you are looking down at you. He was ashamed and embarrassed to be on the side of the gang where he doesn't really belong. He had a scholarship making him easily a Socs, but giving up the chances for his little brother might of came back to him. That probably made the reader think Darry is heroic for the sacrifices he made for his brother education and future. On the other hand Paul probably was looking down on him, thinking how much he is suffering. Probably even laughing inside.
Darry feels sad, jealous, and ashamed to be on the other side of his future. Seeing Paul makes him remember his broken future.