I found something interesting. There is actually a clinical trial underway in Texas that is focused on the use of bone-marrow-derived mesenchymal stem cells (delivered intravenously) for the treatment of PD. It is currently recruiting PwPs.

Allogeneic Bone Marrow-Derived Mesenchymal Stem Cell Therapy for Idiopathic Parkinson's Disease - Full Text View - ClinicalTrials.gov

NB: This trial is using bone-marrow-derived MSCs (i.e. not umbilical cord MSCs).

Hopefully it won't be too long before we see an MSCs-for-PD clinical trial that uses umbilical cord MSCs.