Robert Cumming, 26, was one of two men to appear before Mansfield magistrates in connection with the incident in January.
Jake Henderson posted a 50-second video on the social media platform in which he used a racially abusive word to describe Ms Patel following a Covid-19 briefing.
In the video, Henderson could be heard saying: "As a white man, I won't be listening to people of colour."
Cumming admitted sharing the video along with the caption "Haters gonna be hating", followed by four laughing emojis.
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Prosecutor Daniel Church said he believed the video was "motivated by hostility towards members of racial groups".
Henderson, 28, from Newark and Cumming pleaded guilty to a single charge of sending a grossly offensive, indecent, obscene or menacing message or matter via a public communication network.
The court was told the video was recovered by police from the phone of Henderson's partner.
One member of the public was quoted to magistrates by the prosecutor, saying: "People have the right to feel their own feelings but he didn't have the right to show his feelings in this way" and added the video could "incite racial hatred".
A Rotherham Borough councillor said the video had "upset" her, adding: "I feel the comments were targeted at everyone in Rotherham, especially the Asian community."
She added: "The comments are very extreme and were not made in a joking manner.
"They were not off-hand racist comments - it was an ideological view."
They were granted unconditional bail until sentencing on August 12.