  We consider two-dimensional (2D) localized vortical modes in the three-wave
system with the quadratic ($\chi ^{(2)}$) nonlinearity, alias nondegenerate
second-harmonic-generating system, guided by the isotropic harmonic-oscillator
(HO) (alias parabolic) confining potential. In addition to the straightforward
realization in optics, the system models mixed atomic-molecular Bose-Einstein
condensates (BECs). The main issue is stability of the vortex modes, which is
investigated through computation of instability growth rates for eigenmodes of
small perturbations, and by means of direct simulations. The threshold of
parametric instability for single-color beams, represented solely by the second
harmonic (SH) with zero vorticity, is found in an analytical form with the help
of the variational approximation (VA). Trapped states with vorticities $\left(
+1,-1,0\right) $ in the two fundamental-frequency (FF) components and the SH
one [the so-called \textit{hidden-vorticity} (HV) modes] are completely
unstable. Also unstable are \textit{semi-vortices} (SVs), with component
vorticities $% \left( 1,0,1\right) $. However, full vortices, with charges
$\left( 1,1,2\right) $, have a well-defined stability region. Unstable full
vortices feature regions of robust dynamical behavior, where they periodically
split and recombine, keeping their vortical content.
