Can I use the word uncle to mean my parent's uncle? I mean without having to say grand-uncle or great-uncle, in the same way can I use the word cousin to mean my parent's cousin.
Yes, it is common to shorten lengthy-but-precise relationship terms. I always call my great aunts and uncles simply aunt or uncle, unless I'm discussing the particulars of the family tree, in which case the generational indicator is important. Similarly, it's not common to refer to someone as your second cousin or cousin once removed in everyday conversation, you'd usually just call them a cousin without further qualification.
In my experience, "uncle" can include "parent's uncle" or "grandparent's uncle". In my family, my parents' aunts and uncles were known by me as "Aunt Dorothy", "Uncle Michael", etc. Sometimes I've referred to one of them in conversing with someone who doesn't know them and, if it's relevant to note that it's my mother's uncle I'm talking about, I'll probably say "my mother's uncle", but I might just as well say "When I was a kid, we used to have Thanksgiving dinner at my aunt's house" even though it was my mother's aunt.
However, neither Merriam-Webster nor the American Heritage Dictionary provide for this extended meaning, though American Heritage provides, even more extensively, "Used as a form of address for an older man, especially by children." https://ahdictionary.com/word/search.html?q=uncle
An uncle is an uncle. You call them Uncle Mike when talking to him or about him. All my fifteen great nieces and nephews call me Aunt Peggy. When explaining a family tree, or introducing your Uncle then you would say, “I’d like you to meet my Great Uncle Mike.” The little kids certainly just say Uncle Mike. I would only use grand for direct parent lines. Peggy