Why Gyrados isn't part Dragon:
Gyrados was not made a Pokemon with the dragon type as a water-dragon typing in generation 1 would be to over-powered with only one weakness - dragon. In the first generation there was only one dragon move - Dragon Rage - which will always only inflict 40 damage.
Why Gyrados is part flying:
Not being part dragon, Gyrados was made part flying as that was the next best thing. This was likely to be because of Koinobori which are carp-shaped wind socks (like kits) that are flown across Japan on Children's Day (5th May), the last day of Golden Week, in honour of sons for a good future and in the hope that they will grow up healthy and strong. This is relates to the fact that Magikarp has grown healthy and strong.
Why do Kingdra and Palkia have the Water-Dragon typing then?
As the generations progressed, more dragon type moves became available along with more Dragon typed Pokemon being created to get STAB off those moves along with many other Pokemon that could use them. This stopped the Dragon type from being so elite and over-powered so Kindra is not overly strong.
Palkia is however a legendary Pokemon with stats and moves to go with the title so...
... that is the point. Palkia's strength, speed and ability to take hits is supposed to be legendary and over-powering.
Between Palkia and it's counter-part (Dialga), there are only 3 weaknesses: Dragon, Fighting and Ground (this is when they were made - Palkia is now weak to Fairy attacks). Actually, because Dialga has 8 50% resistances, 1 25% resistance and 1 immunity compared to Palkia's 1 50% resistance and 2 25% resistances; Dialga has a better defensive typing than Palkia.
Just remember though, only you can have them on your team in the games and they are in the uber tier in competitive battling with other over-powered Pokemon.