For starts all Regina really did was fight to keep Henry. In the first season she was trying to get Emma out of Storybrooke mainly to keep her from getting Henry. The main reason she hated Emma was because she felt that Emma was going to take her son from her. And the length she went to try to get Henry all to herself is saying something.
I'd also like to bring up two common arguments; that Regina abused Henry and that she yelled at him too much. Starting with the yelling to much thing; I think a lot of that was done to protect him. Like in 'That Still Small Voice' right after that mine shaft started to crumble. Regina had yelled at Henry to stay back but the kid would not listen. I honestly don't blame her for yelling at him to go wait in the car after all he could have gotten hurt. Later on in that episode he went into the mine and almost died like most mothers she actually did not yell at him, she was just relieved to see the boy in one piece. And then in season 2 when he went to go blow up the well in an attempt to get rid of magic. Of course she yelled at him, he was holding a stick of dynamite and a match, that seems pretty reckless. People like to bring up that she yells at him for stupid things like leaving his shoes out for exsample. I don't feel like that's her being particularly mean to him I feel like she is simply a strict mother, who wants to keep him safe, and when he does something that could get him hurt she punishes him. As for the abuse argument; for one we have never even seen her spank him. I doubt she psychically abused him. When Emma asked Archie if Regina would ever hurt Henry, he responded "Oh no not Henry, everyone else maybe, but not Henry."
Season 2 showed so much about her love for Henry; she dropped a good deal of the stricness mentioned above, she even gave him the option to go live with David rather than her, because she knew that's what he wanted. Because she realized she was trapping him like her mother did to her, and she didn't want that. And in the same scene she told him she was willing to redeem herself for him. And she was doing great; she would tuck him in and got him help with the nightmares. She even stopped using magic for him for a period of time. After a while Gold managed to get her to help him cast a curse on that well, in the end--at Henry's request--she went back on that plan and saved Emma and Snow, risking her own life. She tried really hard for Henry and I don't think that the tears she shed after Emma told Henry she murdered Archie were for show.
And then there's season 3 in it's entirety. Everything she's been doing on that island is to get Henry back. And she proved that she'd do anything to get him back. She even started to acknowledge him as Emma's child and not just her own, which was a major problem with their relationship. Regina's unease and impatience to me is a clear sign that she genuinely cares for Henry. The way she told Tinkerbell "with Henry I finally got something right." This whole season has been proof of her love.
And if none of that is proof enough, both Lana herself and Adam&Eddy confirmed that Regina does love Henry.
Now I'm not saying she's a perfect parent, I don't think it was healthy for him when she had him in therapy for believing in the curse nor was it healthy for her to let him be such a loner. But I still don't think she's a bad parent, I just think she has a hard time expressing love and was too protective of the kid.
Long story short; Regina has a hard time expressing love and a twisted why of doing so. But all things considered, I don't think she's a horrid parent and I do think she loves Henry very much.
