gravity is the attraction between two masses. The larger the masses the larger the attraction. It also quarters when doubling the distance between the masses.
Now the sun is very big and heavy, the planets not so much.
The galaxy as a whole is also pretty heavy, but very very far away, especially the gravitational center. So the gravitational forces are pretty small because of the distance and mainly affect the heaviest body, the sun.
So the planets do experience the gravity of the galaxy, but it is so small in comparison to the sun’s gravity, that is not really noticeable.
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