Hopes of a free-flowing encounter were dimmed by the grey, rain-laden skies, that left pools of water on the pitch.
The Dragons' debut-making fly-half Arlidge landed a fifth-minute penalty, and Richard Fussell went close to a try after Arlidge's chip over the top of the defence.
The home side's plans were disrupted by an early leg injury to full-back Ashley Smith, with Connor taking his place.
Parks missed the chance to equalise with a 20th-minute penalty, then made amends with a second chance as the Dragons went offside at a ruck.
New Dragons captain Tom Willis looked comfortable in the quagmire conditions as the home forwards went through the phases, but neither try line was seriously threatened.
Glasgow took the lead for the first time just after the break, when Parks landed a fine long-range penalty when Richard Parks transgressed at a ruck.
The visiting fly-half's influence on the game began to increase, and his kick down the centre of the field won a scrum five when Wayne Evans could not deal with the greasy conditions.
Parks found space from the scrum, feinting left before targeting the blind-side, but a certain try was blown when his pass went too high for Lome Fa'atau to take cleanly.
Glasgow's pressure increased with only a series of desperate tackles holding them out as Parks missed a drop-goal and a further penalty.
As the Dragons looked to have weathered the storm, Connor foolishly tagged Thom Evans as the winger chased a kick.
The summer recruit from the Ospreys was sin-binned and Parks nailed the penalty to take the lead to six points.
The fly-half's fourth penalty was his best, the kick sneaking over from wide on the touch-line to take the Scots two scores clear.
The Dragons had kicked earlier penalty chances to touch, but now they turned back to goal, Connor missing from 40m.
With the game winding down, John Beattie earnt a yellow card for a blatant ruck infringement and Connor landed the penalty from in front of the posts to secure a losing bonus point.
Man-of-the-match Dan Parks: "The conditions were very heavy and very tough, but we knew that coming in - our boots were sodden in the warm-up.
"With all that water on top of the pitch it was difficult to play any sort of flamboyant rugby.
"We've always struggled down here and knew how tough it was going to be, so we're really happy with our defence and with the four points."
Dragons: Smith, Selley, Sidey, Stcherbina, Fussell, Arlidge, W. Evans, Black, Willis, R. Thomas, A. Jones, Charteris, Bearman, Parks, L. Evans. Replacements: Connor for Smith (18), Gustafson for Black (61), Harrison for R. Thomas (72), Webb for Parks (72), Ringer for L. Evans (67). Not Used: Maule, Lewis.
Glasgow: Stortoni, Faatau, M. Evans, Morrison, T. Evans, Parks, McMillan, Va'a, Hall, Low, Barker, Kellock, Brown, Barclay, Beattie. Replacements: Henderson for Faatau (54), Tckachuk for Va'a (16), Palepoi for Barker (16). Not Used: Milligan, Swindall, Gregor, Jackson.
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