Do PHP pages on a server run simultaneously?
Congratulations, you have identified a race condition! :-)
Whether PHP pages run in parallel or one after the other depends on the web server. Typically a web server allocates several threads to handle multiple incoming requests at once. So it may indeed happen that several instances of the same script are run in parallel if two or more users request the same page at the same time. Due to timing and scheduling differences it is unpredictable when each page will execute which action exactly.
Hence for such situations as you describe it is important to program actions in an atomic way, meaning that they either complete in their entirety or not at all. In your case you could use locks, transactions, cleverly formed UPDATE statements, UNIQUE indexes or a number of other techniques that avoid the possibility of two users reserving the same thing.
Yes, in general, without getting into too much detail: PHP scripts are executed simultanously for each request separately.
For making sure the problem you mentioned does not occur, you should probably implement feature of your database management system called "transactions". This way if you do something on the database layer and at the end you will find out the reservation can not happen, all the actions made within transaction will be rolled back.
In addition to transactions you should design your application keeping in mind that the problem you mentioned may occur. Thus you should design your database & application in a way allowing you to 1) shorten the time between "checking" and "reserving" as much as possible, 2) stopping the action if you cannot make reservation, and finally - in case of emergency - 3) identifying which reservation came first and which should be revoked.
Another idea, falling into category of "your application's design", may be something we could call "temporary reservation". That means you can temporarily (eg. for a couple of seconds) lock your reservation if you are about to make reservation. After that you can check if you really can make that reservation and either turn it into permanent reservation or just revoke it. I believe some systems also make longer temporary reservations right after the customer begins the process of reserving his/her places. Then, if the process is successful, the reservation is changed into permanent, but if some specific amount of time passes without success, the reservation can be simply revoked, allowing another customer to begin the process.