[SalesForce] How to share a lightning email template folder

Actually, you can get parentId of the Folder just like this: SELECT ParentId FROM Folder

Here is the code you need to execute to retrieve all folders where email templates are stored:

List<EmailTemplate> templates = [SELECT Id, Name, FolderId FROM EmailTemplate];
Set<Id> foldersIdsToCheck = new Set<Id>();
for (EmailTemplate template_i : templates) {
    foldersIdsToCheck.add(template_i.FolderId);
}

Set<Id> foundFoldersIds = new Set<Id>();
List<Folder> foundFolders = new List<Folder>();

while (foldersIdsToCheck.size() > 0) { //retrieving parent folders until they exist
    List<Folder> folders = [SELECT Id, ParentId FROM Folder WHERE Id IN :foldersIdsToCheck];
    foldersIdsToCheck.clear();
    for (Folder folder_i : folders) {
        if (folder_i.ParentId != null) {
            foldersIdsToCheck.add(folder_i.ParentId);
        }

        if (!foundFoldersIds.contains(folder_i.Id)) {//store found folders to list to easily build tree when all are retrived
            foundFoldersIds.add(folder_i.Id);
            foundFolders.add(folder_i);
        }
    }
}

//TODO: build the tree with email templates and related folders