There is a lack of landmarks for the 75 hiking paths on this mountain, but there is a guide called Footpaths of Mt. Parnitha and map by Road Editions that outline these hikes well.
Here are just two of these:
Mont Parnes – Bafi Refuge: you can take the cable car up to Mont Parnes Hotel and follow the easy 3-km route to the Bafi Refuge. From there on you can go to the Flambouri refuge, the spring of Skipiza, Mola, or the base of the cable car. The starting point is the Mont Parnes Hotel towards the old helipad to reach the footpath. A paved road will eventually lead you to the footpath and run under the paved road, eventually going to the Bafi refuge or back down to the cable car station. Generally the route is not difficult and full of amazing nature. Keep an eye out for the red rhombuses and squares to show you the way.
Thrakomakedones 2 – Falmbouri refuge: This path climbs up sharply in a westerly direction. Once you pass the Spring of Koromilia you can visit the Kioussi Cave and then get to the peak of Flambouraki. After this you will eventually get to the Flambouri refuge.