If Dally's (Dallas Winston's) parents have influenced him, we certainly do not see any evidence of it. The only influence you might say they have is from not being around. Maybe that helped cause Dally to be so angry.
Ponyboy describes Dally as something of a wild person. He is very angry -- at the whole world, and has been in trouble with the police more or less all the time since he was 10.
I suppose that maybe he came to be this way because his parents didn't care about him. But other than that, I don't see any way that the influenced him.
There have been many papers written on nurture versus nature. In Dally's case I believe it could have been a combination of both. Dally, whose real name is Dallas Winston, is a Greaser who uses no grease in his hair. He has blue eyes but they burn with a cold hatred towards the world. He had spent three years of his life living in New York in the bad part of town. He was arrested at the age of ten years. He was the toughest and meanest of the Greasers. The writer states that there was no line that separated Dally when it came to him being a Greaser or a hood. In New York he had actually been a part of the real gangs. Dally had a police record such as being arrested for being drunk, disorderly, lied, cheated, stole, rolled drunks, and jumped small kids.
Dally probably developed his personality as the result of his environment at home and in his community. He obviously did not have much supervision and the book states that Johnny was the only thing that he loved. (page 152)
Dally loses it after Johnny dies which is a strong indicator that Dally really has lost the only person that he trusts and loves. If Dally had a good relationship with his parents, where he felt safe and nurtured, he may still have turned out problematic based on the environment he had lived in when he resided in New York, but he would also have gone to them when he was upset and not ended up causing his own demise.
Even though the book does not state anything about Dally's relationship with his parenst, he fits in with the Greasers who are boys that have to serve as family for one another.