How to dissolve the build-up from a paraffin based oil?
Naphtha, paint thinner or mineral spirits are common names for the type of solvent suggested by @NotNicolau, and should be effective. Caution: all are flammable and could be ignited by a spark from machining, from a motor or from a switch. These solvents are considered only mildly toxic, though.

Chlorinated solvents such as trichloroethylene are less flammable and even more effective at removing this "varnish" (probably not paraffin, which would be greasy but not tacky) but they are more toxic and the fumes are more annoying.

Ventilate the area well both during cleaning and afterwards until all solvent has evaporated and blown away.