I recently took US citizenship and want to sponsor my parent to come here. I am familiar with the process of applying for green card for my parents. But this is the unique situation that they are in right now.

They had a family based green card around 1985 and they came to U.S. with me and my brother for a few months. After a few months they went back to their home country for good because of family obligations. The green card got expired and over the course of years lost.

Then I got married to a US citizen and received green card and then citizenship. My parents got a non-immigrant visa few years ago to visit me.
I am ready to file fresh green card for my parents. I am being told that my parents need to surrender their 'old' green card otherwise they will have problems during green card and citizenship process.

How should we proceed? Thanks in advance for your responses.

You can't surrender what you don't have so you explain the circumstances to the authorities. If this leads to a delay there is nothing you can do.

Permanent residents lose their status when they live outside US for more than 12 months. You can just start a new process... nothing to do with the old cards.