Yes, but it comes down to one specific document: the Judgment and Commitment Order.
That order is the official sentencing document signed by the judge at the time of conviction. It spells out the exact terms of the sentence including whether parole is a possibility. If the Judgment and Commitment Order contains a parole provision, then parole eligibility exists regardless of the inmate's current security classification. Maximum security and Class 2 status reflect how the facility is managing the inmate right now, but they do not override what the sentencing judge put in writing.
If there is no parole provision in that order, then the security classification is irrelevant because parole was never part of the sentence to begin with.
Assuming parole is on the table, the security classification and conduct record become very important when the case actually goes before the parole board. Maximum security status is going to raise questions about behavior and risk level. The board will want to see a pattern of improved conduct, completed programming, and genuine evidence that the person is ready for supervised release. Going in front of the board while still classified at maximum is not impossible, but it requires a stronger case than someone coming in from a lower security level.
The first step is getting eyes on that Judgment and Commitment Order. An attorney or the inmate's case manager can pull it and confirm whether parole is written into the sentence. Everything else follows from that answer.