Strictly reading, pubic lice are an infestation, not a disease.
In the Dungeon Master's Guide (pages 256-257) there is not any game term defining disease; there is a summary explanation and few examples that reflect everyday experiences in the real world.
From a real world point of view, an infestation of parasites could induce a disease, but the infestation itself is not a disease. Anyway, the terminology is not completely clear (see here1 for example), but strictly speaking such kind of parasites may induce diseases, but they are not the disease (see this abstract from a scientific publication).
The DM could rule otherwise
As an example of Rule 0, the DM can decide that what better suits their campaign, the tone of the party and, mainly, what is more fun for the table.
For example, they may decide that the faith of the Paladin is so strong that they can purify all the consequences of the sinful acts of their companions, such as STDs.
In my personal opinion, this could lead to really interesting role play between the group, for example if the Paladin and the Rogue have opposite views about "tavern adventuring time" or even if they are on the same page.
1 Credits to Kirt that pointed this out in the comments.