I went to PHES last month and they told me that they have received 600 over apps for the last 2 DAYS ! . Nevertheless they would still ask you to submit your apps online and they will get back to us. 2 weeks later they wrote an email & told me that they have a huge waiting list & my kids have very slim chance and advise me to look for alternate schools.
Good schools now are very full with huge waiting list. Since you are moving here next month, suggest you to start applying for more schools to give you a higher chance. Here are the list (besides Parkhouse ) of schools with Brit curriculum that you can consider.

1. DESS ( apply only if you have Brit passport )
2. Doha College
3. DBS ( Doha British School Montessori)
4. Newton International School
5. Newton British
6. English Modern School

More expensive ones will be Compass & Sherbourne.

Do take note that schools normally takes QR 200 - QR500 as an application fees per child. Don't get upset when they tell you that the school is full as this is common and also a standard answer. People come & go and quite a number of people leaving during the summer hols & don't come back. Just apply & you may be lucky.

Hope that helps.