No. Strictly speaking, in an orbit, two object orbit their common center of mass, called a barycenter. This center is closer to the center whichever object has more mass. For very large mass differences (such as between Earth and the sun) it’s close enough that we usually just say the less massive thing orbits the more massive thing.
Stars have much more mass than planets. Looking at the limits of what are generally considered stars and generally considered planets: the least massive stars are still about 6 times more massive than the most massive planets (~80 Jupiter masses vs ~13). So in all cases, the center of mass will still be closer to the star.
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