A new Dell laptop would almost certainly ship with Windows 8 and use the Extensible Firmware Interface (aka Unified EFI, or UEFI) firmware rather than the old Basic Input/Output System (BIOS) firmware. EFI-based systems usually use the GUID Partition Table (GPT) rather than the older Master Boot Record (MBR) partitioning scheme that's usually (but not always) used with BIOS.
This is relevant because the instructions to which you've linked assume a BIOS-mode installation with MBR. If you're continuing to use GPT and EFI, the procedures will be completely different! According to this document, Solaris now supports EFI-mode booting, but I can't point you to specific instructions for how to install it in this mode.
Most modern EFIs include a Compatibility Support Module (CSM), which enables the computer to boot in BIOS mode. Thus, you may be able to boot in BIOS mode, but this will be easiest if you configure all your OSes to boot in BIOS mode. In the case of Windows, this will require converting the disk from GPT to MBR. For all OSes, if the disk is currently in GPT form, converting to MBR will require re-installing the boot loader. In most cases, you'll need to go into the firmware setup utility and activate the CSM -- it's usually identified by that name or by the name "legacy support" or something similar. Sometimes it's activated by de-activating "UEFI support" or something similar. (There's nothing remotely standardized about these user interfaces, unfortunately.) If you're wiping everything and re-installing, this boot-mode and partition table juggling isn't a big deal, once you understand what needs to be done. If you want to keep Windows, though, it can be a hassle.
Also, computers that ship with Windows 8 almost always ship with Secure Boot active. Disabling this feature is required to install some OSes. Some Linux distributions support Secure Boot but others don't. I have no idea about Solaris. You should be able to disable Secure Boot in your firmware -- but again, precisely how depends on your firmware's user interface.
If necessary, you can switch between EFI-mode and BIOS-mode boots, but this is often awkward. Most EFIs permit doing it using their built-in boot managers, which are usually dreadful tools; or you can install my rEFInd boot manager, which makes things a bit easier. (With rEFInd, you must uncomment the scanfor line in the refind.conf file and ensure that a suitable BIOS-mode option is added to the line.)
As to partition types, parted and GParted are terrible tools for manipulating partition type codes, because they don't provide an explicit way to do this. These tools will set the type code based on the filesystem type you specify, but if you need to use something else, you're out of luck. Instead, you should use fdisk (for MBR disks) or gdisk (for GPT disks) to set partition type codes in any but the handful of standardized ways that libparted permits. FWIW, IIRC Solaris has abandoned the 0x82 type code in favor of something else, but I don't recall what it's now using instead.
