My kid is good for her father, but HORRIBLE for me.

You have to discipline her when she misbehaves. She has more respect for your husband, probably because she knows he would not tolerate her misbehaving. Try swatting her little butt a few times when she defies you and see if that doesn't make a difference!

She is testing you. And if you give in to her she is going to keep doing it. You need to be very firm with her and take away her favorite stuff when she misbehaves. She needs a major time out. And you need to start saying no to her. No TV, no favorite stuffed animal, no dessert, no nothing.

What does he do differently than you do? Take a lesson.

If that doesn't help you must stay right on top of her. Do not let her get away with a single thing. It is hard and frustrating but you have to be the bad guy.

Kids are aware of and fully understand boundaries. think back to when you were a kid. You were fully aware of those adults - teachers, parents, grandparents, aunts, uncles, etc - that you had to behave for. You knew you had to straighten up and fly right for them cuz they werent going to take any crap from you. YOur dd is no different. She has figured out that daddy has boundaries that she cant budge, but you have gray areas she can tromp on. Time to toughen up the rules and the consequences. You'll have a happier kid, and you'll be much more content with your dd

What is her relationship with him like? Compliance can be fear-based without it being healthy or positive. There isn't anything in your question to indicate a meaningful context for what you describe, (mainly you mention her behavior but even that doesn't suggest any basis for why), so I think it's impossible to assume with any certainty that his parenting is "better" than yours just because she is acting out with you.
A 3-year-old who actively acts out with her mother but NEVER for her father, ALWAYS listens & obeys him, doesn't strike me as evidencing particular normal 3-year-old behavior. It sounds like she has learned that she cannot be authentic about herself or her feelings/reactions in her father's presence. I'm not surprised that she has "her normal self" plus all that got bottled up around her father, PLUS the baggage from being forced to adapt herself in that way, busting out when she's with you.

Here's a related quote from Dr. Gabor Mate's book on ADHD:

Do not mistake acquiescence for voluntary "good behavior."
Compliance does not necessarily mean effective or wise parenting. In cases of divorce or separation, for example, a mother may complain that her child acts out with her but is well behaved with the father. One psychologist's report in a custody case I was asked to review even concluded that the father was the better parent because the young child--a five-year-old with clear signs of hyperactivity--seemed better mannered in the psychologist's office with the father than in the presence of the mother. The mother also reported that the child was especially uncontrollable in the day or two immediately following overnight visits with the father, further proof in the mind of the psychologist of her ineptitude as a parent. This psychologist seemed not to have understood that the child's supposed bad behavior with

the mother really represented a sense of greater security. Since the mother did not treat the child as harshly as the father, who, as he proudly explained, would hit the child's fingers as a method of discipline, the child's counterwill reactions were not suppressed in her presence. On the contrary, the suppressed counterwill built up when the child was with the father erupted with all the greater force in the safety of the mother's care.
The children most at risk for problems later in life are those who feel so threatened that their counterwill falls completely silent. Many a good little boy or good little girl grows up to be a depressed and troubled adult.
(end quote)

The "counterwill" he mentions is the immature individual's automatic resistance, or defensive reflexive opposition to another's will. It's what gets engaged in power struggles. I don't think the mother he mentions is necessarily effective as a parent but

that is not his point. He is just saying that the child's behavior, and increased "acting out," is not evidence of her ineptitude (in contrast to the father's competence), but rather reflects that the child CAN act out with her, is more secure & safe with her.

She would benefit, I'm sure, from a parenting approach that defuses rather than engages the child's counterwill, as would you. As would any parent of a 3-year-old, lol!

A child who is carrying a lot of tension, such as one whose reactions are so thoroughly inhibited & suppressed with one parent, is more likely to overreact & signal her difficulties with her behavior at other times, so parenting in a way that builds connection & avoids practices that tend to entrench oppositionality will be helpful. Parenting in ways that support the attachment bond will help to decrease & de-emphasize natural counterwill. (That's what Mate's book is all about, ways to do that)

A couple days of being "bad cop" will make a huge difference....if u say no mean it...my 3 yr old daughter has been testing me to the MAX!! Even to the point where I've gone to the bathroom and cried....it's exhausting and emotionally draining, but it needs to be done....good luck!! (((hugs))))

It's hard to offer advice without more context. As Dardenella suggests, take a look at what you and your DH do differently. Are you consistent? Do you offer consequences for her behaviors and do you follow through once you've issued them? How do you speak to her? Watch your tone of voice - is it firm and authoritative or is it pleading and wishy-washy?

IF DD is running in the house when you don't want her too, for example. Do you say "Please don't do that. I'll count to 3 and if you're still running you'll go to time out. 1...2...I mean it. You better stop! Ok, now listen..."

OR, is it simply "No running in the house. Stop now or go to your room. Ok, that's it. Room. Now."