Okay, I'm back. Thanks for your patience.
Palmdale Mun. Code Section 41.07 provides in relevant part: "The following accessory uses are permitted in the R-1 zone, provided that such uses are established on the same lot or parcel of land as the principal dwelling unit and that such accessory uses are incidental to and do not substantially alter the character of any permitted principal use...A. Animal keeping, subject to the following provisions. Outdoor animal enclosures shall be located pursuant to Section 30.09.D....1. Dogs and cats, provided that not more than two (2) dogs and two (2) cats over the age of four (4) months may be kept for each dwelling unit."
Section 16.02.D. defines a "dwelling unit" as: "one (1) or more rooms in a building or portion thereof, designed, intended to be used or used for occupancy by one (1) family for living and sleeping quarters, and containing only one (1) kitchen, not including hotels or motels."
The ordinance seems pretty straightforward to me, because it limits a "dwelling unit" to two dogs and two cats over the age of four months.
Since I don't know the layout of your property, I can't say whether or not you have more than one dwelling unit per the definition. However, given that you are apparently zoned R-1, I would assume that there is only a single dwelling unit on the property. If so, then the ordinance appears to me to limit the number of dogs and cats to two animals of each species.
I realize that my answer may not be exactly what you were hoping to read. However, under the circumstances, the best that I can do is to explain what the law is and is not, so that you can avoid expending valuable resources looking for answers that do not exist, and concentrate on the options that are actually available.
I hope I've answered your question. Please let me know if you require further clarification. And, please provide a positive feedback rating for my answer (click 3, 4 or 5 stars) -- otherwise, I receive nothing for my efforts in your behalf.
Thanks again for using Justanswer!