What are the criteria for earning a mustard stain?

In order for an "airborne insertion" to be considered a "combat jump", there are a number of formal criteria that have to be met. While it's not spelled out succinctly in the regulations (specifically AR 600-8-22*), it does spell it out if you follow though the references while jumping around the document.

The one thing it does say clearly is, "Day-to-day combat assault type missions in an already established combat zone do not qualify for award".

Regarding the award of a bronze star ("mustard stain") on the Parachute Badge the key parts are:
---- "A bronze service star is authorized to be worn on Parachutist Badges to denote a Soldier’s participation in a combat parachute jump. Orders are required to confirm award of these badges"
---- "A Soldier’s combat parachute jump credit is tied directly to the assault landing credit decision (para 7–23) for the unit to which the Soldier is attached or assigned at the time of the assault. Should a unit be denied air assault credit, no air assault credit for purpose of this badge will accrue to the individual Soldiers of that unit."

Following through the regulation to para 7-23 (the paragraph head is "assault-landing credit-arrowhead device", but the criteria applies to parachute jumps, amphibious landings, or helicopter assault landings), the relevant info for parachute jumps are:
---- "Make a parachute jump into enemy-held territory as a part of an organized force carrying out an assigned tactical mission."
---- "The assault operation will be of such scope as to warrant DA designation as a battle or campaign and be of such magnitude as to include tactical elements of at least one other Service."

So, in order to receive credit for a "Combat Jump" instead of a "jump into a combat zone", it has to be formalized, it must be Joint, and it has to be a stand-alone "battle or campaign" by the Department of the Army.

Regarding that last one (battle or campaign), there is additional criteria that is used by the theater and HQDA in determining what combat actions might fit into that category:
---- "The forces committed should be spearheading a major assault into enemy-controlled territory."
---- " The operation will be such that the committed forces will ultimately control the area in which they have landed and not rely on immediate link-up with other forces or extrication after a hit-and-run type of mission."

The Senior Army Command (of at least 3-star rank) in the theater's AO can approve what constitutes a "Combat Jump" and submit it to HQDA which then forwards it to the Center for Military History to confirm it meets all the criteria. If it does pass scrutiny, then HRC will publish a General Order with the established Combat Jump credit.

So, while there were many "jumps into combat" there are a lot fewer "Combat Jumps" that Soldiers can wear a bronze service star on the Parachutist Badge.
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* AR 600-8-22 - https://armypubs.army.mil/epubs/DR_pubs/DR_a/pdf/web/ARN18147_R600_8_22_admin2_FINAL.pdf

The 56 combat jumps are listed at https://www.globalsecurity.org/military/ops/airborne-jumps.htm

Not for sure, but has to be a jump into a combat zone. When I was in RVN, the 173rd was lacking in troopers with the stain. Long time ago, but I do remember that it was the 25th Inf Div that secured the drop zone. It was in War Zone D I think. Well covered in the media and flags out watch and the 25th to make sure Charles not cause havoc. 1967 I think, or if not 1966, those years I was in country.