The Parker Solar Probe is intend to reach 600k km/h. With this velocity, reaching the heliopause (123AU - 18 billion kms) would take about 3 years, after accelerating to full speed.
The Parker Solar Probe will achieve this incredibly high velocity by making it go into a highly elliptical orbit. It will do this with gravitational assists from Venus, and it will do this seven times. Each encounter with Venus is designed to make the Parker Solar Probe go even slower at aphelion. After the seventh encounter, the Parker Solar Probe will be going about 1/3 of Venus's orbital velocity at aphelion. This slowness at Venus's orbital altitude is what will result in the Parker Solar Probe's incredibly high velocity at perihelion.
That said, a small velocity change at perihelion could make the Parker Solar Probe escape the solar system were it were equipped with thrusters to do so. At its closest approach, the Parker Solar Probe will be going at 5.969 km/s (21488 km/h) slower than escape velocity from the Sun. A 6 km/s impulsive burn performed at perihelion would put the Parker Solar Probe on a slow escape trajectory, with a $v_\infty$ (velocity after escaping the solar system) of about 3.5 km/s. This is slower than Voyager 1. But thanks to the Oberth effect, a small increase in the magnitude of that impulsive burn would result in a much faster $v_\infty$. For example, a 7 km/s impulsive burn performed at perihelion would make the Parker Solar Probe be on a trajectory with a $v_\infty$ of 20 km/s, faster than Voyager 1.
An altered approach could result in an even higher velocity on escape from the solar system: use Jupiter instead of Venus. Suppose multiple gravity assists from Jupiter are used to make a future probe dive as close to the Sun as the Parker Space Probe is planned to get. Instead of lacking 5.969 km/s to escape the pull of the Sun, this future probe would be within 867 m/s from escaping the solar system at perihelion. Instead of a $v_\infty$ of 20 km/s, a 7 km/s impulsive burn at perihelion with this altered approach would result in a $v_\infty$ of almost 50 km/s. The Oberth effect can yield highly nonintuitive results.