Recent experiments performed in current-driven nanocontacts with strong perpendicular anisotropy have shown that spin-transfer torque can drive self-localized spin waves [W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek, G. E. W. Bauer, and C. Serpico, Phys. Rev. B 81, 014426 (2010); S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, and J. Akerman, Phys. Status Solidi RRL, 5, 432 (2011)], that above a certain intensity threshold can condense into a nanosized and highly nonlinear dynamic state known as a magnetic droplet soliton [S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. Akerman, Science 339, 1295 (2013)]. Here we demonstrate analytically, numerically, and experimentally that at sufficiently large driving currents and for a spin polarization direction tilted away from the normal to a nanocontact plane, the circular droplet soliton can become unstable against the excitations in the form of periodic deformations of its perimeter. We also show that these perimeter excitation modes (PEMs) can be excited parametrically when the fundamental droplet soliton precession frequency is close to the double frequency of one of the PEMs. As a consequence, with increasing magnitude of a bias magnetic field the PEMs with progressively higher indices and frequencies can be excited. Full qualitative and partly quantitative agreement with experiment confirm the presented theoretical picture.