NASA says there are only 5 dwarf planets in the system. But, it's all pretty arbitrary. The line between planet, dwarf planet and asteroid are all pretty fuzzy.

An alien civilization looking at the Sol system might say that it's only got one planet, Jupiter. Everything else is so much smaller that they're not really significant.

Another logical cut-off would be that planets had to be bigger than any moons in the system. If we went by that standard, Jupiter, Saturn, Uranus, Neptune, Earth, Venus and Mars could all still count as planets, but Mercury would get ditched because it's smaller than Ganymede and Titan.

What's funny is that we're still using the name "planet" which comes from "asteres planētai", meaning "wandering star". For the Greeks what mattered wasn't the size or the mass, it was how bright they were. That meant that a tiny object near the sun like Mercury (Hermes) got the name planet, because despite being tiny, the fact it's close to the sun means it reflects a lot of light. And Jupiter (Zeus) and Saturn (Cronus) got named not because they're so big, but because they're big and far away from the sun, which means they reflect sunlight in a similar way to the much smaller inner planets. Earth's moon might have been given the name "planet" if it had been a lot smaller and/or further away.