Does the prostration of thankfulness invalidate the prayer, and is it permissible for the one being led in prayer to part ways with the Imam if the Imam performs a prostration of thankfulness during the prayer?

Prostration in Surat Sad is a matter of scholarly disagreement. The view of the Shafi'is and Hanbalis is that it is a prostration of gratitude (sajdat shukr) that invalidates the prayer. Therefore, it would have been better for you not to separate from the Imam but to refrain from prostrating, wait until he raises his head from prostration, and then follow him. The Shafi'is have explicitly stated that for someone praying behind a Hanafi or someone similar who believes in the validity of the prayer, it is preferable to wait for him until he stands up from his prostration and then follow him, though separating from him is permissible. What you did is permissible according to many scholars, and the issue is a matter subject to ijtihad (independent juristic reasoning).