Changing clause of condition to absolute phrase and participle w

(I) The problem with your first example is that

• (1b) Having money, I'll buy a car.

defaults to the reading/s

• (1a') I have the money; I'll buy a car. or
• (1a'') I have the money, so I'll buy a car.

unless there is context forcing a different timeframe, such as

• (1bb) I'll have made tens of thousands by this time next year. Having money, I'll buy a car.

Standalone, it is unacceptable to reduce your first sentence in this way.

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(II) I'd not use 'allow' without a direct object; I'd choose

• (2a') If my parents allow me to / let me[,] I'll go abroad.

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• (3b) Weather permitting, we'll get there by Thursday.

shows a fully idiomatic use of a near-identical absolute clause to the one you suggest. I'd say, however, that

• (2b') Parents permitting, I'll go abroad next year

is at least partly tongue-in-cheek (so fine in a humorous or rueful conversation) whereas 'Parents allowing, ...', which doesn't mimic the archetypal example so closely, sounds a little unnatural. Not ungrammatical, and not giving an unintended meaning, but I'd find an alternative.