Your in-laws are way out of line. As far as I can tell, you have done nothing wrong. You have the right to set reasonable boundaries around your in-laws’ relationship with your child. The important word is “reasonable.” It is certainly reasonable for you to not want another pet. You’re right that you would end up caring for it. It is not reasonable for your mother-in-law to think she can do and say anything she wants just because she is a grandmother. That entitlement doesn’t come with the job description.
Where is your husband in all of this? You shouldn’t be negotiating boundaries on your own. This is an important developmental milestone in the maturing of your own family. Your husband, their son, needs to be clearly on the same page with you. The two of you – together – need to decide what is in the best interests of your child. You need – together – to have a loving talk with your in-laws. You can certainly tell them that you appreciate their interest in your daughter. You can tell them how important it is for your daughter to have loving grandparents. But you can also tell them that it isn’t appropriate for them not to honor your role as your daughter’s parents. It isn’t at all supportive of your authority as parents if your daughter understands that she can always appeal to a “higher court,” her grandparents. It isn’t healthy for her to grow up with tension between the two generations of people who love her.
When you have this conversation, it’s important that you stay loving and clear, not angry. Engaging in an argument gives the impression that you can be talked out of your position. Supporting each other means staying clear and rational.
I sincerely hope that you and your husband can work together to resolve this situation. Often grandparents like these do back off when they realize that the younger couple really means it when they assert boundaries.
I wish you well.