Myocardial T1 mapping has become increasingly established for tissue characterization in numerous cardiomyopathies. However, the commonly used end-diastolic single-shot imaging imposes restrictions on the spatio-temporal resolution. In this work, we explored increased parallel imaging accelerations and higher resolutions, in conjunction with an image denoising technique that exploits inter-dependencies between the multiple images using random matrix theory. Following parallel imaging reconstruction, common noise characteristics across the images are extracted from the singular value decomposition of a Gaussian random matrix and denoised using locally low-rank regularization. Application of this technique to SAPPHIRE T1 mapping shows no corruption of the T1 time and enables parallel imaging acceleration up to 4 with an in-plane resolution of 1.1x1.1mm2 at clinical image quality.