We present a directed search for continuous gravitational wave (CW) signals emitted by spinning neutron stars located in the inner parsecs of the Galactic Center (GC). Compelling evidence for the presence of a numerous population of neutron stars has been reported in the literature, turning this region into a very interesting place to look for CWs. In this search, data from the full O3 LIGO-Virgo run in the detector frequency band [10,2000] Hz have been used. No significant detection was found and 95% confidence level upper limits on the signal strain amplitude were computed, over the full search band, with the deepest limit of about 7.6×10-26 at ≃142 Hz. These results are significantly more constraining than those reported in previous searches. We use these limits to put constraints on the fiducial neutron star ellipticity and r-mode amplitude. These limits can be also translated into constraints in the black hole mass-boson mass plane for a hypothetical population of boson clouds around spinning black holes located in the GC.