Neutron-rich Pm (Z=61) isotopes were studied by delayed γ-ray spectroscopy at RIBF, RIKEN Nishina Center using the in-flight fission of a 345 MeV/nucleon 238U beam. A cluster-type Ge detector array, EURICA, was used to measure the delayed γ rays from stopped ions. Isomeric γ decays were observed in 159Pm and 161Pm with half-lives of 4.97(12) μs and 0.79(4) μs, respectively. Level schemes for 159Pm and 161Pm were constructed in this study. The isomeric states of 159Pm and 161Pm could be interpreted as two quasiparticle excitations of neutrons with the configurations of ν(7/2633⊗5/2523) and ν(7/2633⊗1/2521), respectively. They are analogous to the isomers that have been observed systematically in other even-mass N=98 and N=100 isotones in this region. A projected shell model calculation was performed and it reproduced the order of three-quasiparticle states only if new Nilsson parameters with an N-dependent spin-orbit interaction were used. This work demonstrates that the strength of spin-orbit interactions in standard Nilsson parameters needs to be modified to study the properties of neutron-rich rare-earth nuclei around A=165, and provides new evidence supporting the existence of the deformed N=98 subshell gap in odd-mass nuclei for the first time.