Can married daughter can have rights on her parents property

Since the property is on your mom's name, you do not have any legal right's on the said property. You can convince your parents to give you a share in the sale proceedings to buy a new property for yourself. 

Learned Mr. Vishnu has given valuable and practical advice. Kindly follow it. The son {your brother} wants to be away in other town and wants have support of his {your} parents.

The parents can decline to sell the house or give share to son and daughter.

If parents are willing the settlement by your parents in their life time is the best option.

It is a wish that relations should remain sweet but sometimes relation turn sour.

The lady {your mother} is the owner of property thru her husband {your father} and both are alive. It is felt that the parents are at their liberty to give any amount to any one.

If parents wish they can give equal share to son and daughter.

The succession and matter of right to estate is after the death of the owner.

Hindu Succession Law: Female dying without will; first right is upon sons and daughters including children of any pre-deceased son or daughter, and the husband.

Property acquired from husband: first right with heirs of husband in case of absence of  his sons or daughters.