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We present the experimental demonstration of the parallel parametric generation of spin-waves in a microscaled yttrium iron garnet waveguide with nanoscale thickness. Using Brillouin light scattering microscopy, we observe the excitation of the first and second waveguide modes generated by a stripline microwave pumping source. Micromagnetic simulations reveal the wave vector of the parametrically generated spin-waves. Based on analytical calculations, which are in excellent agreement with our experiments and simulations, we prove that the spin-wave radiation losses are the determinative term of the parametric instability threshold in this miniaturized system. The used method enables the direct excitation and amplification of nanometer spin-waves dominated by exchange interactions. Our results pave the way for integrated magnonics based on insulating nano-magnets.