Australia is a signatory to and a supporter of the UN Declaration of Human Rights but should this fact committ the State and other organs of Government as well as the ordinary citizens to supporting it?

Is it possible ethically to accept parts of it, at our convenience, and to reject others?

In terms of the state, I think that signing and ratifying a treaty does create a moral obligation to respect and uphold it. If Australia no longer wishes to respect its treaty obligations, then it should open withdraw from them, rather than simply ignoring them.

In terms of the individual, the position is different. You are free to disagree with, criticise or oppose any act of the government, including the rarification of a treaty. I think most people would agree that in some circumstances there is a moral right, or even a moral duty, of civil disobedience to acts of the government.

I don’t think that the question of civil disobedience really arises with respect to the UN Declaration of Human Rights, but I certainly don’t see any reason why an Australian citizen can’t morally disagree with the Convention, or aspects of it, merely because Australia has ratified it. (Their stance might be morally objectionable on other grounds, but not on that ground.)

What's suggested is that treaties be signed and then ignored, at your convenience, so you can say you support truth, justice, and the U.N. way of life without actually supporting it.

No, I think that's unethical. Quite common but unethical. In the U.S., the President will, with great fanfare suppot and sign something knowing full well all treaties have to be approved by the Senate. Then the Senate votes it down, which the President may delight the President.