Adam Idah headed the hosts into the lead before Adam Lang equalised shortly before half-time.
Parrott’s late winner, where he capitalised on a defensive error to race through on goal and slot home, saw O’Shea celebrate with players and staff in the dugout.
“The situation I’m in – I’m born in Waterford and I’m managing Ireland. We won a game in my third game in charge,” the 43-year-old added.
“We have played Belgium and Switzerland, two really good teams, and now we have played a team who were unbeaten in 14 games. I’m super proud of all the staff, the backroom staff and, most importantly, the players.
“They put in the dedication and application. Yeah, we should be better and can be better, and we will be better.”
O’Shea said he “knew the type of form” Parrott was in after the 22-year-old netted a hat-trick in Excelsior Rotterdam’s 4-1 win over NAC Breda.
“I spoke about how everyone was going to be needed, especially at this time of year because it is a tricky period for a team.
“A bit more detail on our passing in the first half and we would have really opened Hungary up.”
The Republic of Ireland travel to Portugal on 11 June in their final match before their Nations League opener against England in September.
O’Shea’s future beyond the game in Aveiro is unclear, but he says he wants to “enjoy” the win over Hungary before turning his attentions to Portugal.
“They are probably one of the favourites for the Euros and they are going to be in the mix for it.
“We’re back wining a game of football, let’s really soak it in but we’ll go through it all and then prepare for Portugal.”