That's wrong. The green pilot lights on the dash should only ever flash with the turn signals. They aren't to light up with the parking lights, or light up steadily under any other condition.
Have you double-checked to make sure you're using the correct kind of bulbs in the front turn signals? If you accidentally put BA15s-base bulbs (single-contact/single-filament, like an 1156) instead of BAY15d-base bulbs (two-filament/double-contact, like an 1157) you'll bridge the two contacts in the socket base and one of the symptoms is turn signal pilot lights lighting up steadily with the first click of the light switch. Shouldn't be possible to make this bulb swap, because the index pins are different, but sloppy sockets and/or bulbs make it possible. Same can happen with "LED bulbs", most of which are unsafe junk, even if they have the nominally correct base type.