AC start up causes lights on separate circuits to blink

Tyson's comment could be it, if so, he should post it as an answer. I'll list off a collection of questions and reasons why it could matter.

What year was the house built? Are you running the unit on the same wall outlet you have been for 6 years? What else have you added to the home in the last 6 year, and how much of it is on currently?

Where I'm going with this is a few areas:

1.) If the house has aluminum wiring, # 2 is more likely

2.) Running an A/C of that size is taxing a typical 15 Amp circuit (assuming this is on a typical 15 Amp circuit), over years of "abuse" connections degrade especially with aluminum. The wall outlet/wiring/breaker can be getting hot and drawing more current. It wouldn't hurt to secure power and check connections

3.) If you've added other computers, TV's other A/C's, other appliances, etc, they all draw power, it can add to the load to the panel, thus impacting incandescent bulbs on the same panel

4.) did you add a Hot Tub? Any other NEW breakers to your panel? (still part of #3)

5.) Ensure the only thing running on the same breaker as the A/C is the A/C.

6.) Turn off things that are not needed and run a test. Helps to rule out most of 1-5.

7.) Don't use extension cords. Even though appliance cords will work, it will still cause more voltage drop over a longer run than the wall outlet *(thus, more heat)

8.) What kind of Bulbs are flickering? Are these different bulbs than last year? They might be more 'sensitive'

9.) Per Tyson's comment, have you had any severe windstorms in the last season? it could absolutely be on the pole side if so, or in the meter as noted

10.) As commented by Ed Beal regarding the starter cap, the unit itself can be wearing out. Again, if a starter cap is bad the unit typically wont start and you may see a flicker when it tries, if it is going bad but marginal, you can see flickering due to transients on the line while the compressor is starting. Also, along these lines, the compressor itself will wear out over time. (enamel breaks down, lubrication breaks down) even a dirty filter (for air or for coolant) can increase the load on the unit resulting in more power being drawn.

If it's not what @nybman pointed out. See below.

Is your dwelling power being fed with a service drop? I have run over this situation twice in the last few years and both times the neutral was broken on the service drop. Do what @tyson said call the utility company and see if they can check the continuity on the neutral from the xfmr to the meter. It's their responsibility to maintain the drop.

PS Don't be surprised if you put a voltmeter on the phase and neutral and get 120V. You will because it is still connected to the ground. That is what may be happening, the neutral is feeding to the panel, then through the ground, to the butt ground at the pole and back to the xfmr. The resistance in that path will cause the lights to dim when a large circuit surge from the AC pulls the current down.