The fastest way to reach Piatra Mare Peak (1844 m) is to start from Timișu de Sus and continue upwards by Cascada Tamina (Tamina waterfall). It's a 3.5 - 4 hour hike up to the peak and 2.5 - 3 hours back.
If you're driving on DN1 from Predeal to Brașov, you can leave your car in the parking lot on the right of the road, just after you've exited Timișu de Sus. The trail opens on the right hand side with a small bridge leading to a forestry road marked with a blue stripe.
The blue stripe trail leads to Cascada Tamina. The waterfall is one hour away, and you can hike through it (there are steel stairs installed inside the canyon formed by the river).
To hike through the waterfall leave the trail when you see the waterfall sign, descend on the right side until you reach the bottom of the waterfall and then climb your way up using the steel stairs. Once you've exited the waterfall, climb back to where you've started from and re-enter the trail towards Piatra Mare.
The trail continues through the forest, intersecting the forestry road you've started from, until you hit the alpine region at Stâna Pietricica.
The trail then follows the red cross mark () and then the red stripe () towards the peak.
Google Maps version of the trail above: Timișu de Sus - Piatra Mare Map
The KML is loadable in Google Earth and the visualization is pretty spectacular: