mission [...] acceptée par la simple réception

20:18 Jan 18, 2019 

there is already quite a lot of context, more than enough:

Law/Patents - Law: Contract(s) / Real estate

payment (for the purchased property) through an intermediary acting as "le tiers convenu" in plain speak "a mutually agreed third party that will act as intermediary for the payment" - we don't know why there are doing this but knowing it wouldn't change one iota the translation.

there is also the "implied context" called the basics of contract law - a contract can by accepted "by conduct" - which is the case here - and the "conduct" showing acceptance of this specific contract is O_N_L_Y the taking of the monies - passing the monies to the final recipient is NOT part of "accepting" the contract but of "executing" the contract.

In fact it's more than implied, it's even explicitly said in the ST:

"La mission du tiers convenu
(1) sera acceptée par la simple réception des fonds et
(2) exécutée par la remise de ces fonds à son bénéficiaire."

Calling this relatively simple service rendered to the Seller and Buyer of some real estate a "project" would sound a bit ... over the top? grandiose?

17:24 Jan 18, 2019 

That was a mistake that I have since corrected, though I cannot correct the header to my answer.

16:46 Jan 18, 2019 

Receipt as in the act of receiving.

22:44 Jan 17, 2019 

How about this: the mere reception of the funds will constitute the acceptance of the third party task and be fulfilled by the delivery of the funds to its beneficiary

21:50 Jan 17, 2019 

I think more context would help here. We don't know what the "fonds" or the "tiers" are, or what is going on. "Mission" is rarely a good translation for "mission", and the sentence needs a subject, verb, and object, unlike the French.
Depending on the context, you could say something like "When the third party receives the funds and transfers them to its beneficiary, it will be deemed to have accepted the project".

20:17 Jan 17, 2019 

"Acceptance of the third party mission agreed upon will be [confirmed] by receipt of the funds and performed by transfer/remittance of the funds to the beneficiary".
Or:
"The third party mission agreed upon will be accepted by receipt of the funds and ..."
Or:
"Receipt of the funds will be considered as acceptance...."

20:10 Jan 17, 2019 

In this case, "par" --> "by", in the sense of "through", "by means of".