"Anyone know the story behind the Qatar A320 parked at INV"
They're no stranger to INV, seen every year The Qatari royals own a sporting estate at Inverinate near Kyle of Lochalsh, for which INV is the nearest airport. As noted above, they are believed to own other estates in the Highlands too.

Not quite, its Sheikh Mohammed bin Rashid Al Maktoum, the ruler of UAE, that owns the Inverinate Estate (one of the largest in Scotland). The Qatari's do own a mansion on the private island of Eilean Aigas just west of Inverness which maybe explains the Qatar plane being in INV. When the Sheikh is in town it is usually pretty obvious due to a pristine G5 and a couple of Dubai 737's parked on the north apron (for his support staff). Given the political tensions between Qatar and UAE I doubt there is much interaction between the two families, even when in Scotland.