This work proposes MP-Grasp4D (magnetization-prepared golden-angle radial sparse parallel 4D) MRI, a free-breathing, inversion recovery (IR)-prepared, time-resolved 4D MRI technique with improved T1-weighted contrast. MP-Grasp4D MRI acquisition incorporates IR preparation into a radial gradient echo sequence. MP-Grasp4D employs a golden-angle navi-stack-of-stars sampling scheme, where imaging data of rotating radial stacks and navigator stacks (acquired at a consistent rotation angle) are alternately acquired. The navigator stacks are used to estimate a temporal basis for low-rank subspace-constrained reconstruction. This allows for the simultaneous capture of both IR-induced contrast changes and respiratory motion. One temporal frame of the imaging volume in MP-Grasp4D MRI is reconstructed from a single stack and an adjacent navigator stack on average, resulting in a nominal temporal resolution of 0.16 seconds per volume. Images corresponding to the optimal inversion time (TI) can be retrospectively selected for providing the best image contrast. Reader studies were conducted to assess the performance of MP-Grasp4D MRI in liver imaging across 30 subjects in comparison with standard Grasp4D MRI without IR preparation. MP-Grasp4D MRI received significantly higher scores (P < 0.05) than Grasp4D in all assessment categories. There was a moderate to almost perfect agreement (kappa coefficient from 0.42 to 0.9) between the two readers for image quality assessment. When the scan time is reduced, MP-Grasp4D MRI preserves image contrast and quality, demonstrating additional acceleration capability. MP-Grasp4D MRI improves T1-weighted contrast for free-breathing time-resolved 4D MRI and eliminates the need for explicit motion compensation. This method is expected to be valuable in different MRI applications such as MR-guided radiotherapy.