Parrott struck in injury-time as Ireland won 3-2 on Sunday to book their place in the World Cup playoffs.
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Afterwards, the former Tottenham striker said: "I am really emotional right now. What a night.
"This is why we love football, things like this can happen. I love where I am from and it means the world to me.
"My family are here, and this is the first time I have cried in years! I really can't believe it. I am really emotional right now. What a night."
Parrott also said: "I said against Portugal that this is what dreams are made of, but I think tonight, I will never have a better night in my whole life.
"It is a fairytale. You can't even dream about something like that.
"I have no words to describe the emotions right now."