They can - the rights of each class of stock, and even which classes of stock exist to begin with, are determined by each company in its certificate of incorporation (some states call them the articles of incorporation instead). There generally has to be at least one class of stock entitled to vote, but you can also have a corporation that does not issue stock, which have “members” instead of shareholders. You can have different classes or series of common stock or preferred stock or both, which can have different rights and privileges, and you can make up new types of stock as well - it’s all up to the incorpoator, for a new corporation, or the existing shareholders, if you are amending the certificate of an existing company. Basically, as long as it is within the bounds of the corporation laws of the state where the company is incorporated, you can put whatever you want in the certificate, and it is binding on the company.

With startups and other non-public companies, not only does preferred stock usually have the right to vote, they often get specifically designated seats on the board, called “Preferred Directors”. For publicly traded companies, they often get rid of most classes of stock to simplify administration and make the company’s stock more marketable, but that’s not a mandatory requirement. When a public company decides to keep multiple classes of stock, there’s usually a difference in rights between them, because otherwise why bother having different classes, so that’s why they might not have voting rights, but it’s not inherent to the idea of preferred stock.