The country routinely celebrates big anniversaries, like our bicentennial, so I don't really see the issue here. Whatever partisan politics people are seeing will be gone in a few years, but we will always have footage of the Army being celebrated for its 250th birthday. I don't find patriotism cringey. Could the money be more effective elsewhere? Sure, but you could always say that. We send billions to other countries without a dime in return, so I have no problem with money going towards this. We shouldn't put a price tag on celebrating the people who defend our country.
I'd rather Trump not tie his birthday into the parade, but Trump is gonna Trump. It's not going to take away from the importance of the event for me. Like I said, I don't doubt that you and many you served with think there is no need for it. But plenty of people who did serve do, along with millions of appreciative citizens.