In paragraph 100 of his official report covering the operations on Morris Island, Brigadier-General Quincy Gillmore noted:
On July 18,  immediately after our second repulse at Fort Wagner, orders were given to convert the position occupied by our right batteries on the 18th of July, and named Battery Reynolds, into a strong defensive line, capable of resisting a formidable sortie. From that time this line took the name of the “first parallel.”
Looking to Major Thomas Brooks’ map of the operations, here is the first parallel that Gillmore mentioned:
So what is a parallel? Let’s turn to that American military theorist Dennis Hart Mahan (page 146, A Treatise on Field Fortification):
The parallel is a long line of trench, concentric with, or parallel to, the works of the point of attack, which it envelopes…
So rather simple, a parallel is a counter-work of fortification used by a besieging force as a line of operations. Mahan went on to describe other works in relation to the parallel:
… and the boyaux are simply communications, in a zigzag direction, which lead to the parallel. The parallels and the boyaux both serve as a covered way, in which the troops can circulate with safety; but the boyaux are used exclusively for this purpose, whereas the parallels serve to contain troops who retrain the excursions of the garrison, and protect the workmen whilst carrying forward the approaches.
Brooks’ map shows these boyaux in relation to the parallel – both front and rear of the first parallel. The wording “covered way” applied to the works on Morris Island. Troops moved from encampments well to the rear of the lines up to the parallel and beyond inside a system of trenches, remaining out of direct sight (mostly) of the Confederates on Battery Wagner.
Notice the nature of the “zig-zags” of the boyaux. The lengths to the left (west) are slightly off parallel to Battery Wagner. But as they run to the right (east, towards the beach) the angle in relation to the Confederate works is much sharper, and the length is twice that of the left side runs. Neither lengths in the zig-zags allow a direct flank shot by the Confederates. And those running right respect the Confederate defenses inland of Morris Island. That’s a covered way.
In his manual, which all those building works at Morris Island were familiar with, went on to describe the proper placement of parallels:
The position of the First Parallel is six hundred yards from the most advanced salients; it should embrace within its extent the faces of all the collateral works that protect the point of attack.
At the distance of six hundred yards, the fire of the work is not very troublesome; and it is within good range for a ricochet fire, if it be found necessary to erect the enfilading batteries near the first parallel.
Sounds good, but the first parallel on Morris Island stood 1,335 yards from Battery Wagner. Over twice the distance recommended by Mahan! When Mahan wrote his treatise in 1852, long range columbiads were still new, evolving weapons and rifled guns were experimental novelties. By 1862, breeching batteries could back off 1,200 yards from the enemy works. Likewise, the enemy could make work at 600 yard parallel very troublesome.
On July 19, 1863, Brooks noted in his journal actions taken to strengthen the first parallel against Battery Wagner:
1. Extend the obstacle each way; on the right by a return along the beach; on the left by a boom across the creek, to obstruct the passage of the enemy’s small boats.
2. Place Requa batteries in position on the flanks of this parallel, for its defense.
3. Elevate platforms and rebuild revetting and embrasures for six guns on the right of the parallel.
4. Transform the center light gun battery into a siege mortar battery.
5. Extend the siege gun battery, so that it may contain one additional 30-pounder Parrott rifle.
6. Build one bomb-proof magazine for the service of the armament of the first-parallel.
7. Build emplacements and a bomb-proof magazine for four 10-inch seacoast mortars in rear of first parallel, opposite the Beacon House.
These work instructions resulted in the structures seen on the map above. Note the Requa battery position on the left flank of the parallel. Another went up on the left in front of the parallel.
The “siege” of Battery Wagner was getting under way. But there’s a twist here. Gillmore now realized he could combine phases 2 and 3 of his original plan as his men worked from Morris Island. The siege operations would extend to include Fort Sumter at ranges considered only theoretically possible before the Civil War.
(Aside from Mahan’s Treatise (linked above), Citations from OR, Series I, Volume 28, Part I, Serial 46, pages 17, 272-3.)