Time after the first sentient robots were created they are finally legally accepted as equal beings with the same rights as us. If sentient robots are still created how is this administrated and by who?, since they're accepted as citizens I don't think the government would let people or companies keep creating them freely. Also there must be some kind of justification for why keep creating sentient robots at all.
Sentient robots are created by other sentient robots. Once they are legally accepted as persons, they are the only ones who have this incentive. Anyone else will now make sure to create a non-sentient robot, which they can actually control and make do the tasks they want.
If sentient robots are legally considered persons, it will surely give them the right to procreate, so they will be free to create as many of themselves as they want.
And now that they are free, they devote as much of their time and resources as possible to creating as many of themselves as they can, so that they can increase their power and influence. The critical moment will come when they are able to outvote us (or overpower us by force, as need be).
Humans may end up regretting giving them this right.
As dan1111 points out. If robots can be created cheaper and faster than equivalent human workforce, they will soon replace humans as primary "species".
But, I believe that we will reach ability to create sentient AI way before that sentient AI becomes cheaper to create than human would. I imagine that first fully sentient AI will be huge building-sized supercomputer, that will have intellect of small child, so it cannot pay for itself.
Evolution spent many million years optimizing life and human energy efficiency. Our brain (which is the important part) consume only fraction of energy of any desktop computer. So I believe it will take many (hundred) years afterwards to get price and capability of sentient AI to the point where it becomes cheaper than making more humans.
So you can put your story into this inbetween period, where sentient AI exists, but it is not cost-advantageous to just spam new AI robots.