Context. Theory and observations indicate that the distribution of stars along the horizontal branch of Galactic globular clusters mainly depends on the metal content. However, the existence of globular clusters with similar metal content and absolute age but different horizontal branch morphologies, suggests the presence of another parameter affecting the star distribution along the branch. Aims: To investigate the variation of the horizontal branch morphology in Galactic globular clusters, we define a new photometric horizontal branch morphology index, overcoming some of the limitations and degeneracies affecting similar indices available in the literature. Methods: We took advantage of a sample of 64 Galactic globular clusters, with both space-based imaging data (Advanced Camera for Surveys survey of Galactic globular clusters) and homogeneous ground-based photometric catalogues in five different bands (U, B, V, R, I). The new index, τHB, is defined as the ratio between the areas subtended by the cumulative number distribution in magnitude (I) and in colour (V - I) of all stars along the horizontal branch. Results: This new index shows a linear trend over the entire range in metallicity (-2.35 ≤ [Fe/H] ≤ -0.12) covered by our Galactic globular cluster sample. We found a linear relation between τHB and absolute cluster ages. We also found a quadratic anti-correlation with [Fe/H], becoming linear when we eliminate the age effect on τHB values. Moreover, we identified a subsample of eight clusters that are peculiar according to their τHB values. These clusters have bluer horizontal branch morphology when compared to typical ones of similar metallicity. These findings allow us to define them as the 'second parameter' clusters in the sample. A comparison with synthetic horizontal branch models suggests that they cannot be entirely explained with a spread in helium content.