Although end-to-end robot learning has shown some success for robot manipulation, the learned policies are often not sufficiently robust to variations in object pose or geometry. To improve the policy generalization, we introduce spatially-grounded parameterized motion primitives in our method HACMan++. Specifically, we propose an action representation consisting of three components: what primitive type (such as grasp or push) to execute, where the primitive will be grounded (e.g. where the gripper will make contact with the world), and how the primitive motion is executed, such as parameters specifying the push direction or grasp orientation. These three components define a novel discrete-continuous action space for reinforcement learning. Our framework enables robot agents to learn to chain diverse motion primitives together and select appropriate primitive parameters to complete long-horizon manipulation tasks. By grounding the primitives on a spatial location in the environment, our method is able to effectively generalize across object shape and pose variations. Our approach significantly outperforms existing methods, particularly in complex scenarios demanding both high-level sequential reasoning and object generalization. With zero-shot sim-to-real transfer, our policy succeeds in challenging real-world manipulation tasks, with generalization to unseen objects. Videos can be found on the project website: https://sgmp-rss2024.github.io.