Tourism victories really depend on a lot of factors and, for me playing on Emperor, a turn 324 victory is entirely reasonable, especially since Peter is the in game. To win a tourism victory, you basically need to have more international tourists than the civ with the most domestic tourists. So, you needed 335 international tourists to win the game. That's a lot. Looking at what you provided, four national parks isn't enough. They are really good a generating tourism in the long turn so you want to get as many as you can as early as you can. More Seaside Resorts would also be good but those are hard to get. Once you get Conservation, and the Radio, you basically need to rework your empire to fit as many parks in resorts in it as possible. Rock Bands also need to be used strategically, specifically spending at the civs with the most tourism since Rock Bands take away domestic tourists and therefore reduce the amount of tourists you need to win the game.