No, they should not.
There actually IS a trend, primarily in Europe (for the moment) to grant apes human rights. This is actually part of a larger campaign which denies that humans are better than animals, and which also denies that humans are in any way unique in God’s creation.
The people pushing this agenda would have us believe that humans are animals, an accident of evolution and not a work of the Creator. In fact, in their view there is no God.
They deny God because then there are no such pesky things as moral laws to limit our desires and restrict our “freedom.”
And they will tell you that there is no such thing as “human dignity” endowed to us by God merely because we are human. Their “human dignity” is endowed by the State based on who you are and what you can do (for the state - or for them). If you are less than they, then you must sacrifice for them - up to and including your life. If you are less than they, you become a “useless eater”, as the Nazis called them.
Beware this group and others like them.