I have heard that the mitzvah to honor ones parents includes a commandment to honor one's oldest brother. Is this really the case?
The Talmud (Ketubot 103a) states:
כבד את אביך ואת אמך את אביך זו אשת אביך ואת אמך זו בעל אמך וי"ו יתירה לרבות את אחיך הגדול
Honor your father [et avikha] and your mother [ve’et immekha]. The preposition et in the phrase: Your father; this teaches that you must honor your father’s wife. Similarly, the preposition et in the phrase: And your mother; this teaches that you must honor your mother’s husband. And the extra letter vav, which is appended as a prefix in the phrase “ve’et immekha” is included in order to add your older brother to those who must be honored.
Rambam codifies this in Hilkhot Mamrim (6:17):
ומדברי סופרים, שיהיה אדם חייב בכבוד אחיו הגדול ככבוד אביו
By rabbinic teaching one is obligated to honour his elder brother, as he honours his father.
This is in turn quoted by the Shulhan Arukh (YD 240:22).
It should be noted that there is a dispute whether this refers exclusively to the oldest brother, or to any older brother. See Gilyon Maharsha to Shulhan Arukh (YD 240:22) who cites the the view that it refers only to the oldest brother, while others such as the Arizal cited by Hida in Pnei David (V'zot Haberakha: 13) opine that it applies to any older brother.