I met some People who run A Bike Hire Service in the Phoenix Park and this is their Logo on their Leaflets and their Web Site is
They are situated just inside the Main Gate ,just go through the Pillars of the Park Gate on Pargate Street and Conyngham Road Junction across the Road from the Bus Garage.
To Book a Bike,Tel 086-26 56 258
Open 7 Days a Week Weather Depending. you are not just confined to the Phoenix Park but can Tour the Tourists Sites of the City as well.
I have added their Web Site at the Botton of my Blog Page on the Left,for your convenience if you are interested in them.