Dolphins deserve rights, some experts say, because they have consciousness and are persons. This view of persons ends rights, it doesn't expand rights.
According to some experts who support a Declaration of Rights for Cetaceans, dolphins have enough intelligence to count as persons, and persons deserve rights. Dolphins, according to their studies, are self-aware individuals. This function counts them among those "persons" who deserve rights.
This view of persons and rights is called "functionalism." Essentially, it's the idea that if someone or something meets the criteria of whatever functions count, then they're persons and deserve rights. This view of human beings is what is often used to discount unborn humans as non-persons with no rights. It has even been applied to those at the end of life who no longer have the minimum functions, thereby ending their right to life.
This view leads to problems explained here. Personhood becomes a degreed kind of thing. The logical consequence is that some persons deserve more rights than others since they have greater functions. Also, whatever list of functions are given as the requirement for personhood, you can find counter-examples of people we'd want to count as persons but who don't fulfill the required functions.
Functionalism just can't provide adequate grounding for rights.
Further, nature doesn't ground rights. You can't get an ought (rights) from an is (nature). If persons have value that deserves protection of their rights, then you can't get that from nature or evolution. There are no values in nature - just facts. You can't expand rights from natural properties - in fact, you lose all rights. There are no rights if nature is all there is.
Most animal rights activits are evolutionists. If our origin has no meaning, and our final destiny has no meaning, you can't conjure up meaning and value in between. The worldview expunges all rights, it can't support expanding them.
Functionalism can't ground rights. And nature can't accrue rights. There is no adquate claim to give rights to dolphins.