We propose a morphological classification based on the parsec scale structure of fifty-one High Frequency Peakers (HFPs) from the "bright" HFP sample. VLBA images at two adjacent frequencies (chosen among 8.4, 15.3, 22.2 and 43.2GHz) have been used to investigate the morphological properties of the HFPs in the optically thin part of their spectrum. We confirm that there is quite a clear distinction between the pc-scale radio structure of galaxies and quasars: the 78% of the galaxies show a "Double/Triple" morphology, typical of Compact Symmetric Objects (CSOs), while the 87% of the quasars are characterised by Core-Jet or unresolved structure. This suggests that most HFP candidates identified with quasars are likely blazar objects in which a flaring self-absorbed component at the jet base was outshining the remainder of the source at the time of the selection based on the spectral shape. Among the sources classified as CSOs or candidates it is possible to find extremely young radio sources with ages of about 100 years or even less.