When Charlie Parker goes to Dizzy Gillespie's house in the middle of the night and asks Dizzy to write down a tune, the year is 1953. The tune is "Now's the Time", published and recorded in 1945.
Despite the great deal of attention paid to the music of Charlie Parker, authenticity was not strictly adhered to in all respects of the musical performance. The guitarist in the rock and roll show was playing a contemporary Ibanez RG series electric guitar - a modern high-quality Japanese brand that was popular among shredders such as Joe Satriani. Period instruments would have certainly been available.
At the Jewish wedding, the first shot of the orchestra shows both horns playing, but only the trumpet is heard. The saxophone joins later on.
When Bird dictates the telegram to be sent to Chan, it's printed as he speaks. He says "Forgive me for not being in the hospital with you," but the telegram reads "Forgive me for not being there with you while you were at the hospital."