It is annoying that the paper is behind a paywall.

What seems to be going on is that it is known that the tyrosine hydroxylase gene contains a CRE element. The CRE element binds a protein called CREB; this is what they showed is enhanced by aspirin. CREB is a transcription factor; it activates the tyrosine hydroxylase gene, leading to production of more tyrosine hydroxylase protein in the nigra and hence more L-DOPA there as in your scheme.

Whether this works in people as well as mice remains to be seen. Also, I don't know enough about the pharmacology of exogenous/therapeutic L-DOPA to know to what extent its transport into the brain is rate-limiting.

I wish I had known about this before. An uncle of mine (in NZ) died from PD. I would have suggested low-dose aspirin to him.