Yes packages to hunt within National Parks are available. Some are trophy packages and some are meat hunts aimed more at local hunters.
Some Provinces will sell them at auction(you or somebody representing you needs to be at the auction and you bid as per any other auction, highest bidder wins) and some Provinces sell them by tender(blind auction), so you look at what is offered in the package, decide what you are prepared to pay and then put in a price and wit to find out who gets the hunt.
Typically these hunts are then sold over a specific time period. Mostly accommodation is available within the National park and although basic in most camps, is more than adequate.
You do not get free reign to hunt anywhere in the park, you only hunt designated areas and are always accompanied by National Parks game scout.
It takes some planning but have done quite a few of these hunts with clients. In some cases, depending on the Game warden you may be permitted to harvest other animals that where not normally part of the package eg. Blue wildebeest, Impala, Warthog, Zebra, Giraffe, Black wildebeest, Blesbuck etc. This is however not a guarantee and will depend entirely on the Game warden, who will know what the animals are they want to reduce numbers for. These are typically animals that they would have culled themselves.
Con's they are typically animal and time specific.
Pro's they are more affordable and you have a good chance of of getting better than average trophies. Even the meat hunts are more affordable.
I will be bidding on some hunts again this year.