According to the datasheet[1], both S25FL129P0 (256kB sectors) and
S25FL129P1 (64kB sectors) support dual read, quad read, dual i/o read
and quad i/o read.
I have verified dual read to be working for S25FL129P1 on a dual
capable spi controller. Quad as well as S25FL129P0 is untested, lacking
hardware to verify.