Can I pay from the HSA account for medical expenses for my parents whom I'm not claiming on my taxes?

Not unless you can claim them as dependents.

Whose Medical Expenses Can You Include? 

You can generally include medical expenses you pay for yourself, as well as those you pay for someone who was your spouse or your dependent either when the services were provided or when you paid for them. There are different rules for decedents and for individuals who are the subject of multiple support agreements. See Support claimed under a multiple support agreement, later, under Qualifying Relative. 

Qualifying Relative

A qualifying relative is a person: 1. Who is your: a. Son, daughter, stepchild, or foster child, or a descendant of any of them (for example, your grandchild), b. Brother, sister, half brother, half sister, or a son or daughter of any of them, c. Father, mother, or an ancestor or sibling of either of them (for example, your grandmother, grandfather, aunt, or uncle), d. Stepbrother, stepsister, stepfather, stepmother, son-in-law, daughter-in-law, father-in-law, mother-in-law, brother-in-law, or sister-in-law, or e. Any other person (other than your spouse) who lived with you all year as a member of your household if your relationship didn't violate local law, 2. Who wasn't a qualifying child (see Qualifying Child, earlier) of any taxpayer for 2017, and 3. For whom you provided over half of the support in 2017. But see Child of divorced or separated parents, earlier, Support claimed under a multiple support agreement, next, and Kidnapped child under Qualifying Relative in Pub. 501.

https://www.irs.gov/pub/irs-pdf/p502.pdf