The possibility that Sumerians originated from north Mesopotamia raise the probability that Proto-Sumerians had the occasion to contact the Fertile Crescent farmers who moved to South Caucasus and further north of Caucasus. This could mean that some lexical parallels between Sumerian and PIE are genuine. Aleksi Sahala from the Helsinki institute made a summary of proposed parallels. Not all cases are of good quality but some have high chances to be real cognates.
Sum. gud, gu4 'ox, bull; cattle'. ~ PIE *gwou(s)- 'cow; ox'; Hitt. *kuṷāu-;
Skr. go ( ), Gk. bous (βοῦς); TochB keŭ; ON. *kú, Arm. kov
Cattle was domesticated in West Asia core Fertile Crescent region. From there it moved to various regions including the South and North Caucasus. It is not hard to imagine that the term used by pastoralists in Pontic and Caspian steppe was from North Mesopotamia as Sumerians themselves. The Egyptian word "ka" ox, "kaut" cow can have the same origin. Its remarkable that farmers of ancient Egypt had T1a1a and H2, haplotypes, which was present in historic Armenia/South Caucasus farmers also.
Sum. šáḫ(a) 'pig; boar' → Akk. šaḫû 'pig'; Ug. šeḫû 'pig' ~ PIE *suh1-
'swine'; Skr. sūkara (सक ); TochB suwo; Lat. sūs; Goth. swein.
Variants: šaḫ, šúḫ? (ŠUBUR). The reading with <a> is more widely accepted and supported by the Akkadian correspondent. Similar word is also found from Kartvelian languages, GZ ešw- 'wild boar, pig'. All these words probably share a common prehistoric etymology.
Possibly another Fertile Crescent term with similar history. The presence in Kartvelian is also interesting. The Armenian "xoz" is not directly derived from PIE but could be related to same arealic word or from Parthian.
Sum. sí-sí 'horse'

? Akk. sisium 'horse' ← Hurr. issi(a) 'horse' ~ PIE
*h1ekwos 'horse'; Hitt. aśuwas; CLuw a-aš-šu; Skt. áśva (अश); PIA. *aĉwa-;
Lat. equus; Alb. sasë, Arm. eš
The trajectory of this word is without much doubt from PIE to Sumerian. The author conjecture about the Indo-Iranian mediation but the Armenian "eš / išoy" which was meaning horse in the past is the best source of Hurrian form (Petrosyan 2002). We describe the paleogenomic rational for this trajectory in Petrosyan and Palyan 2023. The Armenian word initially meaning horse had a semantic change later. See Martirosyan 2009 for this change.
Sumerians and Akkadians got the word from Hurrians. The bulk of Sumerian attestations are from URIII period which is a post Akkadian period.
Sum. urud(a) 'copper' → Akk. erû 'copper' ~ PIE *h1reudh
-ó- 'red'; Skt.
rudhira, Av. raoðita; TochA rtär; Gk. eruthros (ἐρυθρός); Lith.
raudonas; Gaul roudos; ON rjóðr;
Another remarkable word. The semantic shift is probably from color name to metal term like in the case of "argentum" silver. This word is attested in Sumerian from 4th millenium BC. Copper use is older in Highlands than in Mesopotamia. If the word for color red was initially an IE word then the most likely period is the Chaff faced ware dated at Late Chalcolithic (4200-3500BCE). This period was mix and almost certainly had IE speakers among them.
For other linguistic parallels see the link in the comments.
See also related subject PIE - Semitic lexical parallels.