We have established a method to measure picograms-per-gram (pg g$^{-1}$) levels of $^{238}$U and $^{232}$Th in scintillation films by combining the dry ashing method and inductively coupled plasma mass spectrometry. Trace amounts of $^{238}$U and $^{232}$Th were measured in up to 2~g of the scintillation film with almost 100% collection efficiency. This paper details the experimental procedure, including the pretreatment of the samples and labware, detection limit of the method, collection efficiencies of $^{238}$U and $^{232}$Th, and measurement of $^{238}$U and $^{232}$Th in a polyethylene naphthalate film. This method is also applicable to $^{238}$U and $^{232}$Th measurements in other low-background organic materials for rare event search experiments.