Internet service providers (ISPs) are the owners of the infrastructures through which you perform all your online surfing. They indirectly have access to all your data and online browsing histories with your location inclusive. Possibly, they can see what you are doing on the site you visited, which they can store for future reference or targeted advertising.
Their monitoring access is limited to encrypted sites, and Youtube is one of them. Hypertext Transfer Protocol Secure (HTTPS) secures your connection between your online device and the website you are visiting from dubious attackers and limits what your service providers can see.
When you visit the Youtube site, your ISP cannot see exactly which video you are watching. All they do is take a record of how much time you spend on the Youtube site and your visiting intervals. They can also dictate the number of bandwidth you use while watching Youtube videos.
Taking note of all these, there is a possible path your ISP can take to know the actual video you are watching, especially when compelled by law enforcement agencies to do so. They can use Deep Packet Inspection (DPI) to dictate the exact video you watched on Youtube, proving that your privacy while browsing HTTPS sites is not hundred percent guaranteed.