Yes, you can.
Because PRS is a private school and does not receive any public funding, there is nothing to prevent enrolling in both. Many children enroll in a homeschool enrichment program while enrolled in PRS with no problem using their statement of enrollment in place of an NOI. You can print or download a copy of your statement of enrollment from your account page. Keep in mind that many public schools and districts may not understand what an umbrella school is. You are welcome to contact PRS if you need assistance.