your front left parking light isn't properly installed...
If front and back both "light" up and it's only the left side blinking rapidly it's not the relay it's just the bulb. Like Sultan said, the bulb may not be seated in the socket properly or wiring may me loose somewhere. Try twitching the bulb and wires around until it gets steady again.
It's the relay if the left and right side are both "lit" and blinking rapidly.
Since you did change all the bulbs they should all light up. Since it's only happening on the left side of your turn signals and you know for sure they are brand new bulbs that is probably the area you are going to have to fix. The left side. Make sure all your bulbs are lighting up or this won't help.
Last make sure you have the correct wattage on the bulbs that can cause them to flicker faster too