A primary design goal of companion social robots is to increase a user's perceived level of companionship. One way to examine perceived companionship is through parasocial presence. In the scope of human-technology interactions, parasocial presence can be defined as the extent to which the technology promotes, whether as a medium or by itself, the perception of understanding, social connection, engagement, and intimacy. One specific aspect of parasocial relationships which has been explored in human-robot interaction is para-friendship, or the sense of self-disclosure and social support a person perceives in a parasocial relationship. The present research examined how the relationship between a user's personality and the perceived personality of a companion social robot influences the relationship of intimacy, understanding, enjoyability, and involvement on para-friendship. This research also investigated whether the user and robot's personalities affect para-friendship and the intention to use the robot as a social companion. Due to their designed goal of serving as social companions, understanding the relationship of a user's personality and perceived robot personality on parasocial presence, parasocial friendship, and use intention is one important link to improve Human-Robot Interaction design (HRI) and increase the marketability of social robots. Results showed that personality similarity predicted parasocial presence, but may not be as critical as perceived robot personality. Additionally, while parasocial presence predicted parasocial friendship, this effect was limited for parasocial involvement. Finally, higher feelings of parasocial presence promoted greater use intention. Implications for robot development, personality aspects of robot friendship formation, and opinions of human-robot relationships are discussed.