From your description, my guess is that you aren't getting the broadcast stations directly, you're getting Spectrum's retransmission of them. That would mean that your TV is set to receive cable channels rather than broadcast and that Spectrum isn't encrypting the locals in this area yet (but they can if they want to, so don't expect it to last).
I don't know if Spectrum still has analog channels. Did your parents' old TV have a digital tuner? The manual for the digital converter/DVR you got them doesn't indicate that it has an analog tuner but apparently it can tune unencrypted cable channels. On the Channel Search page, set Signal Type to Cable instead of Air and rescan. I won't get into the issue of whether it's legal or ethical to use cable-supplied signals without paying for them.
As a test of where you're getting your channels from, see if you get any that are not broadcast such as public, educational or government access channels (WMHT doesn't count).
If you want to get the local channels with an antenna, I'd start with an unamplified rabbit ears with loop for about $10-$15. A small panel antenna might be OK for UHF channels but would probably have trouble with high VHF (WXXA, WNYT and WNYA) and would be useless for low VHF (WRGB).