https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(21)00451-5/fulltext https://www.thelancet.com/action/showPdf?pii=S1473-3099(21)00451-5 (pdf link)
Table 1 shows that the expected number of unrelated "background" cases during an equivalent period in 2020 was about 26+/-3 per 100,000. The inactivated vaccine (CoronaVac) is believed to have caused be associated with an additional 4.8 cases per 100,000, while the mRNA (Pfizer) have caused is thought to be associated with additional 2 cases per 100,000. In other words, for every person who is affected by the vaccine, 5-10 times as many people will coincidentally develop Bell's Palsy for reasons having nothing to do with vaccination.
The effect size is very small. It is also important to note that the investigation compared two different vaccine technologies for which data were readily available -- Sinovac (inactivated virus) and Pfizer (mRNA) It did not compare Sinovac or Pfizer against other vaccines of the same types. Other vaccines use the same inactivated virus tech as Sinovac (for rabies, influenza, Hep A, etc.), and possible increased incidence of Bell's Palsy for these diseases and vaccines have been investigated elsewhere.