I would like to expand a bit on what johnf has already explained, maybe we can sort it out this way:
If we talk about virtualisation we usually understand it the way it is done in i.e. Linux: you start with a (physical) machine, where you install some OS. In this OS you start some program, which emulates a some virtual hardware. On this hardware you install another OS and start an application on top of that.
With this picture in mind you probably asked why the OS of the system matters - you could run, say, CentOS 5 in the physical system, start VMWare in it and run CentOS 4 as a "guest system".
Virtualization in p-Series systems is different from that concept, because the base OS (which would run on the physical machine above) and the emulation program are missing - their functions are done in the systems hardware directly. A p-Series system is a big box with a lot of processors, memory, adapters and disks. There is also a small system connected to this box which manages it. This is called a "Hardware Management Console" or HMC. In the HMC (a small and highly customized Linux system) you select a certain number of resources - processors, memory, adapters, .... - and give this to a "LPAR": a "Logical PARtition", which is a definition of a virtual machine. This virtual machine can be booted, installed, shut down, like a physical system.
When this system was invented by IBM roughly 10 years ago it quickly transpired that there are some sorts of resources which are hard to manage: if you give each system a disk (which it needs to boot) you have to install a lot of disks in your big box - and the same is true for network adapters. Especially so, because these two types of resources are limiting factors: every machine has to have at least one disk to boot from and a network adapter to communicate with the world outside. Either you install some spare adapters and disks or you can't just create and fire up another LPAR when you need one.
This is why they invented "VIOS", the "Virtual I/O Server". This is a special kind of LPAR, which gets all the physical disks and network adapters in the big box. Additionally the VIOS usually get some FC adapter(s) to attach big LUNs from a SAN storage. The VIOS runs some special application which creates virtual network adapters (via the physical ones) and virtual disks (on this SAN storage), which it dedicates to other LPARs. The only physical disks inside the p-Series necessary are the disks the VIOS(es) need to boot from, every other LPAR boots from a "virtual disk", which is a file on the VIOS - quite like the "disk" of a VMWare virtual machine is a file on its hosts filesystem.
If you want to fire up a new LPAR you just create a boot disk and a virtual network adapter on the VIOS, use the HMC to create a LPAR definition where you dedicate (along with memory and processors, which are still real) this virtual disk and virtual network adapter to it and then install it like a normal system. The whole system is booted by first booting the big box itself, then start up the VIOS(es) (there can be several for high-availablility purposes), then start up all the other LPARs.
(If you have questions about this just ask - i have tried to explain it in the most non-technical terms possible, which neither means the concept is easy to understand nor is the explanation rich on details.)
To answer your questions: Yes, you can install AIX 5.3 in LPARs in even the newest p-Series systems you can buy. These systems come without any preinstalled OS anyway because they are in most cases used not as a whole (this would be possible, but is in ractice never done) but as hosts for an indefinite number of LPARs anyway.
You may even - depending on some details in your environment i don't know - be able to "live-migrate" your LPARs from one system to another, which means as much as moving them from one big box to another big box without even shutting them down, complete with their LPAR definitions, their disk resources, network resources, etc..
This does not necessarily mean that it is a wise idea to stay with AIX 5.3 - but neither does it mean it is a wise idea to migrate to 6.x! What is the best here depends on a lot of details nobody can decide over an internet forum. Talk with your technicians and if they are worth their money then stick with their advice.
On one hand AIX 5.3 was by a wide margin the best AIX since 4.3.3 and it will surely be around for several years from now, even if it is not marketed any more by IBM. There will be fixes for a long time to come and the installed base is very big.
On the other hand 6.1 is not bad (and a lot more stable than i.e. 5.1 was) and offers abilities in the virtualisation department that 5.3 lacks. It will also be a lot longer around than 5.3.
I hope this helps.
bakunin