Nathan Lyon was one of three Australia bowlers to take three wickets on day two as England were all out for 258.
But England responded well on day three, having closed the previous day on 30-1.
The hosts reduced Australia to 80-4 before rain meant the final two sessions could not be played.
Play had to be abandoned again in the second Test after a washout on day one.
On his debut, Jofra Archer took his first Test wicket with Chris Woakes and Stuart Broad also adding one apiece.
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Steve Smith, however, remains on 13 not out as England trail by 178 runs.
The weather forecasts Saturday and Sunday mean play should not be disrupted by more rain.
But England face a battle in order to pull back the series to 1-1.
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Story of Day Three
Going into the day 30/1, Cameron Bancroft and Usman Khawaja added 30 runs in eight overs for the Australians on Friday morning.
Jofra Archer removed Bancroft with his first ever Test wicket before Chris Woakes found the edge on Khawaja for Jonny Bairstow to catch.
That had Australia at 60/3 with Stuart Broad then trapping Travis Head lbw.
The tourists improved to 80/4 before lunch but rain washed out the final two sessions of the day.
Steve Smith ended the day 13 not out.
Story of Day Two
Play got underway on time with Australia winning the toss and electing to field.
Josh Hazlewood dismissed Jason Roy for a duck with Joe Root also going cheaply.
Rory Burns impressed again with a half-century while Jonny Bairstow also raised his bat for 50.
Joe Denly and Chris Woakes both got solid starts without going on as England were all out for 258.
Stuart Broad then got an early breakthrough for England, clean bowling David Warner.
But Cameron Bancroft and Usman Khawaja saw Australia through with Archer wicketless from his first six overs in Test cricket.
Story of Day One
Rain washed out the entire day though there had been some glimmer of hope when the umpires inspected and ordered the toss.
The rain came back but not before Jofra Archer had been handed his cap by Chris Jordan.