If they are double contact bulbs, which the #68 is, then that is good. The dimly lit bulb indicates that something is grounding through the lighting circuit. The interior bulbs usually use a 15 amp SFE fuse, and that fuse also controls the tail lamps. This is the next place to look, and look for a bulb that might have a burned filament that is laying across one of the filament stalks. I have seen this happen, and create all sorts of havoc trying to find a problem. What does worry me, is why is this happening when the car is parked, because all circuits should be closed. I am wondering if it might have something to do with your alternator, not being correctly wired. If you disconnect the negative battery cable, and hook a test light between the negative battery cable and the negative post on the battery, it shouldn't illuminate. If it does, then that indicates that there is a draw on the system. The only thing that should draw any power, is the clock, if your car is so equipped, and that draw is only intermittent when the clock mechanism points close to wind the clock. This should happen about 2 seconds every 3 or 4 minutes, so I doubt that is the issue. If the test light illuminates, then I suggest that you remove each fuse until you find the circuit that the draw is on. If none of this fuse removal stops the draw, then it is on the engine side of the car. Most likely the alternator or the alternator regulator. I am going to make the assumption that the car doesn't have an under hood light, but if it does, then remove that bulb also. You can also have a grounding issue, where a circuit is searching for a ground through another circuit. I have seen this happen with incorrectly installed aftermarket radios. An example would be, you step on the brake, and the radio starts to play. Does the car have a power antenna, and is it working, or is it broken. If it doesn't work, then disconnect the power lead to it. Isolating circuits is the only way that you will figure this out. Once you know what the circuit is that is causing the draw, then you can start tracing the wires to find where the problem is located.
I only had one electrical problem that I was never able to resolve, but it wasn't for a lack of trying. Usually, I find that it is a poor ground, so I always start by finding all the grounding wires, and clean and tighten them first. If it is in the braking / taillight circuit, I use a test lead to find the housing that has the poor ground, and correct that. Same with parking lights. These are usually the most troublesome circuits, and the easiest to repair, since it is a matter of cleaning and tightening to re-establish good grounding.
The last thing that might be the problem, is the headlamp switch, but usually they are very "black and white", it either works or it doesn't. No grey area usually.
Check out all the above, and then report back. Also, keep in mind, that you will not be the first person to have a defective "new" battery. The battery in my 2016 Ford had a weak cell, that was wreaking havoc with the electronics in the car, giving me all sort of weird messages on the On Board Diagnostics, until the battery was replaced under the warranty.
