With Greyhound, you can get to many areas near National Parks, but not directly into the parks.
For example in W.A., you can get to Kalbarri or Exmouth via Greyhound (be prepared for very long bus rides), then from these points you can take tours or shuttles into the National Parks.
Because these bus rides are so long, there is also the chance of meeting other travelers who might be interested in sharing the coast of hiring a car to get to the national parks (I did this in Exmouth).
On Oz Experience, you do go to some national parks, but these are more gentle walks than hiking expeditions. For better hiking experiences, I would either join a more local tour or find other travelers to hike with, which with patience is easy to do at the hostels (check the notice boards). For example, to hike in Lamington National Park (not too far from Brisbane), you should be able to arrange with a local tour company to drop you off at Binna Burra and pick you up the next day at O'Reilley's, or vice versa.
It is very easy to travel around Oz. There are countless tour companies wanting your business, and there are always notice boards in hostels full of advertisements for people wanting to do ride-shares or looking for hiking buddies. Always check out visitor info centres, because they can often tell you the best way to get onto the trails! They'll often know about any car shuttles for hikers or tour companies that run hikes.
Another option: Budget a few one-way tours. They're a great way to see the remote areas of the country, as they get you from point A to B and you actually get to do stuff along the way.
If you're traveling in W.A. (Western Aus), Western Exposure does do 3 days of hiking in Karijini, and a day in Kalbarri National Park, both of which are fantastic. There are several companies that do the Gibb River Road and the Kimberly region (I did Wilderness Adventures, which took me from Broome to Darwin on a 9-day trip).
In the Outback, there are heaps of companies all over the Northern Territory that do various safaris, and taking Adventure Tours down from Darwin to Alice is not a bad way to see spots like the Devils Marbles and Katherine Gorge (which you don't get to see via Greyhound). Actually, I'm pretty sure there is now a company in Alice Springs that runs hikes on the Larapinta trail...see if you get any Google results for that. I think there is something similar for the Bibbulmun Track in southwest WA.
Down in Tassie, Under Down Under and Adventure Tours, as well as heaps of smaller companies, also hike in various National Parks. Actually, since hiking is so big in Tassie, I am sure you will easily be able to find some hiking buddies to do some overnight hikes.
I would highly recommend looking at which areas you want to hike (Bibbulmun track, Larapinta, Overland Track, Queensland's Great Walks, etc), then narrow down your internet search to include those park/track names, and key words such as "hiking", and "tour" or "shuttle". One of those "Walking in Australia" books might come in handy too, giving you transport options to and from the parks.
Feel free to message me if you have more hiking in Oz questions. My partner is an avid Aussie hiker