In short, yes. Darth Vader can also read minds like Kylo Ren.
This has been shown in The Return of the Jedi.
When there is fight between Darth Vader and Luke on Death Star in front of Emperor, Darth Vader says this,
VADER: Give yourself to the Dark Side. It is the only way you can save your friends. Yes, your thoughts betray you. Your feelings for them are strong. Especially for... sister. So, you have a twin sister. Your feelings have now betrayed her, too. Obi-Wan was wise to hide her from me. Now his failure is complete. If you will not turn to the Dark Side... then perhaps she will...
Here he was able to read Luke's mind. Though, it appears to me that Kylo Ren has greater skills to read minds.