What does "my late parents" mean?

The adjective late usually means “after the expected time,” or “near the end of a time period,” as in these examples: 

1. I’m sorry I’m late. [=I’m sorry that I arrived after I was supposed to arrive.]
2. I made a late credit card payment. [=I paid after it was due.] 
3. My son was a late talker. [=He started talking at an older age than most children.] 
4. He was born in the late 19th century. [=He was born near the end of the 19th century.]

However, as shown in your sentence (“I am the only living son of my late parents”), late can also mean “not living” or "dead." 

Here are some other sentences in which late means dead. 

1. My late grandmother always wore a hat when she went out. 
2. He made a donation in memory of his late wife. 

Notice that when you want to use the word late to mean dead, it must appear directly before the noun that it describes. The sentence “My grandmother is late” does not mean that she is dead. It means that she is arriving after the time she was supposed to arrive.