Judging by the title I assume you want to rego her MH&PE? Yes, you can rego her even if you don't know her DOB or parents' names, but it costs you more. You will need to age her by her teeth and get a good guesstimate (preferably get a vet's/equine dentist's opinion too) and write it on the application as "approx 1998" or "~1998" and they will put her DOB down as 1 August. IMHR also still accept regos with unknown pedigree, albeit at a higher cost.
My stallion (who is actually AMPS) doesn't have his dam on file as her paperwork was out of date when he was registered, so he just has his sire's name on there and dam is down as "unrecorded", so yes, if AMPS do it, I would say MH&PE will allow you to have only one parent on the papers too.