3C 345 is one of the archetypical active galactic nuclei, showing structural and flux variability on parsec scales near a compact unresolved radio core. During the last 2 years, the source has been undergoing a period of high activity visible in the broad spectral range, from radio through high-energy bands. We have been monitoring parsec-scale radio emission in 3C 345 during this period at monthly intervals, using the VLBA at 15, 24, and 43 GHz. Our radio observations are compared with gamma-ray emission detected by Fermi-LAT in the region including 3C 345 (1FGL J1642.5+3947). Three distinct gamma-ray events observed in this region are associated with the propagation of relativistic plasma condensations inside the radio jet of 3C 345. We report on evidence for the gamma-rays to be produced in a region of the jet of up to 40 pc (de-projected) in extent. This suggests the synchrotron self-Compton process as the most likely mechanism for production of gamma-rays in the source.