The title, Padre de Familia, doesn't directly translate to Family Guy, it roughly translates to "Family Father" of "Father of Family". This is the name of the show in Latin America and this episode references Peter's country of origin and the mostly Mexican content throughout the episode.
It's revealed that Peter didn't know what the September 11, 2001 terrorist attacks were until 2004, or at least the severity.
The children of Carter's Mexican employees are in possession of Ball in a Cup, Mexico’s favorite toy for over 340 years. It is the toy's second appearance, the first being "The Fat Guy Strangler".
This episode shows that Herbert used to be a Corporal in the military.
In "And the Wiener is...", Peter's life flashed before his eyes and there was a clip of his mother holding him in the hospital but shown in this episode as born in Mexico.
The Griffins celebrate Cinco De Mayo with the other illegal immigrants after celebrating Veteran's Day at the beginning. Cinco De Mayo, happening in May and Veteran's Day in November alludes to some time passing during Peter's campaign against immigrants.
Despite Peter being born in Mexico, his mother is a US citizen which would give him the same status. According to American law, Peter Griffin would still be an American even if he was born in Mexico on account of the fact that his mother was an American adult. US law states that if a child is born overseas to one or two parents who are Americans at least 19 years of age, the child is American by birth.