Symbolism, Imagery, Allegory
Both the Old Man and Eddie seem to want to live in their own little alternate realities, which they can shape or define however they please. The Old Man seems to be the true champ of this tendency, taking his delusions to a pretty high level—so high, in fact, that he thinks it's totally legit to claim that he's married to famous songstress Barbara Mandrell.
He makes this clear in an early conversation with Eddie, where he points at an imaginary picture of Barbara Mandrell on the wall of the room:
THE OLD MAN: Ya' know who that is?
EDDIE: I'm not sure.
THE OLD MAN: Barbara Mandrell. That's who that is. Barbara Mandrell. You heard a' her?
EDDIE: Sure.
THE OLD MAN: Well, would you believe me if I told ya' I was married to her?
EDDIE: (pause) No.
THE OLD MAN: Well, see, now that's the difference right there. That's realism. I am actually married to Barbara Mandrell in my mind. Can you understand that?
EDDIE: Sure.
THE OLD MAN: Good. I'm glad we have an understanding. (118-126)
In this moment, the Old Man is basically asking Eddie to buy into his notion that he can qualify whatever nonsense goes on in his mind as reality.
This might seem harmless enough initially—ha ha! This crazy (and imaginary) old man thinks he's married to Barbara Mandrell—how quaint!—but the Old Man's belief that he can just define reality as he pleases goes well beyond fantasizing about country stars.
Later in the play, the Old Man gets super upset when May and Eddie mention that Eddie's mother committed suicide after the Old Man abandoned her, and that is so not something the Old Man wants to hear.
He tries to get Eddie to stop May from talking about how the Old Man hurt people with his bigamy, but Eddie adds on to her story, confirming that his mother committed suicide after the Old Man left for good (and that she used the Old Man's gun). The Old Man is not pleased and rails against Eddie for breaking their "pact."
By that, he apparently means the "understanding" he and Eddie forged when he got Eddie to say he understood the Old Man's need to believe that he was married to Barbara Mandrell. In short, he wants permission to live in his own little world without intrusion from the unpleasant realities he fled from in the first place.
He holds on to that desire right until the play's closing moments, when he reiterates that the woman in the imaginary picture of Barbara Mandrell on the wall is "his":
"Ya' see that picture over there? Ya' see that? Ya' know who that is? That's the woman of my dreams. That's who that is. And she's mine. She's all mine. Forever." (584)
Did you get that? Not only is Mandrell his (imaginary) wife, but she's his. The message is clear, Like his son Eddie, the Old Man is a controlling fantasist who wants to construct his own reality out of wishes, regardless of what actual reality is, and the picture of Mandrell becomes a symbol of his tendencies in this regard—and, by extension, Eddie's, too.