Interesting that it affects lights that are not on the same circuit. FYI, there is a separate forum on this board for Electrical Wiring. You may have better luck posting there.

It is doubtful that there is a real problem. I once lived where the power company transformer on the pole was too small for the load. The lights dimmed slightly in time with the washer agitator oscillations.
Motors at the instant of starting draw several times the running amperage. High amperage draw causes voltage drop. The "LRA" on the nameplate is the Locked Rotor Amps and is the same as the instantaneous draw when starting without added load. The washer mechanism plus the wet clothes is a substantial added load.
Use a plug-in voltmeter, available on eBay, in the receptacle for the washer and observe it while the washer is in use.

Call your power company. They may have a program where they test your circuits to find loose connections(which is what your problem sounds like. A loose neutral or ground can cause that situation.

What do you mean by "starts up", when the valves open to fill, when it starts to agitate or when it starts to spin?

If nothing else has changed, your washing machine motor may be going out or, more likely, the brakes sticking. LIsten carefully to the washing machine. Does it sound any different?

Thanks for the responses. It dims lights when it starts to agitate I believe. It doesn't sound any different than it used to but it is 10 years old.

I can call the power company but you would think that the lights would also dim when the central ac started and it doesn't.

I will also try on the electrical forum. Thanks guys!

"you would think that the lights would also dim when the central ac started and it doesn't. "

The central AC is normally a 240 v load.

If the POCO (or your power drop) had a problem ALL the circuits the house would be affected when the AC started.

Unless you want to learn how an 'Edison circuit' (AKA 'split phase') operates it is NOT as simple as you seem to believe.

Get it checked.

I used to have all my lights throb with my vintage induction cooktop (took a lot of juice to run that thing). We didn't notice it with other appliances but they didn't have that constant on-off that makes it really obvious.

Eventually we had a little electrical fire. Forgot what it was. A bad ground in the circuit box or something simple. But yeah, fire.

So just get it checked.

A bad ground would not cause a problem since grounds only conduct power when there is a fault.

A bad neutral (ASA the 'grounded conductor') can cause real problems.

Neutral issues almost always exhibit as brightening of some lights an dimming of others as the center point of the 120/240 V system moves around as there is not 120 V on each leg.

Update:
Ran washing machine on a different circuit and house lights still dimmed and flickered. Machine does seem to struggle a bit during agitation cycle. Time to call an appliance repair person.

There is some good diagnostic information on appliance parts web sites. Appliancepartspros is good and so is mangemylife.com.