No I don't believe they are the same school. Here is why. You had two school's that had separate football histories combining together to start something completely new and separate. Yes Joplin High School kept the Eagle mascot of Memorial HS, but combined the color schemes of both Parkwood and Memorial.
Here are some other examples of consolidated school's.
Mercy High School (Bulldogs) located in University City consolidated with St. Thomas Aquinas (Falcons) of Florissant in 1985 to form St. Thomas Aquinas-Mercy High School (Falcons)in Florissant. In 2003 St. Thomas Aquinas-Mercy High School and Rosary High School (all girls Catholic school) consolidated to form Trinty Catholic High School (Titans) of St. Louis.
St. Mary's High School (Trojans) Independence and O'Hara High School (Celtics) Kansas City consolidated to form St. Michael the Archangel High School (Guardians) in Lee's Summit.
Here are some examples where consolidation happened, but the history of the program didn't change.
Bishop Hogan Catholic High School (Rams) Kansas City became Hogan Prep Academy Charter School (Rams) Kansas City.
St. Louis County-Day School (Rams) consolidated with Mary Institute (all girls school) to form MICDS (Rams).
Pembroke County-Day School (Red Raiders) consolidated with Sunset Hill (all girls school) to form Pembroke Hill School (Red Raiders).