How to update picklist using REST API

As the other answer says, you can do this via the Tooling API if you want to use a REST call. You can find the details on their answer. The other option is to use the Metadata API which is a SOAP API. This stackexchange post has some example code about updating picklists using the Metadata API. The first link to the Metadata API docs should get you started on how to use it. If you are trying to do this in Apex, there is an apex wrapper for the Metadata API on Github.

There is an idea for implementing the Metadata API in Apex natively here, and it appears that the latest update on that idea indicates that as of October 2019 they are not planning on updating that API in the near future.

Now, It can be done using REST calls with help of Tooling API.

The above given solutions works perfectly using Apex Metadata Service (https://andyinthecloud.com/2013/10/27/introduction-to-calling-the-metadata-api-from-apex/); but it requires to import all code for even a small requirement. After, release of Tooling API, I find it lifesaver for such implementation without importing and managing too much code. We just need to make a callout using Tooling API.

Full detailed article is here: https://sfdcian.com/update-picklist-value-and-add-it-in-record-type-using-apex-salesforce/

PATCH
/services/data/v41.0/tooling/sobjects/CustomField/00N540000071QJG

Set metadata as body

{"Metadata" : {
    "businessOwnerGroup" : null,
    "businessOwnerUser" : null,
    "businessStatus" : null,
    "caseSensitive" : null,
    "complianceGroup" : null,
    "customDataType" : null,
    "defaultValue" : null,
    "deleteConstraint" : null,
    "deprecated" : null,
    "description" : null,
    "displayFormat" : null,
    "displayLocationInDecimal" : null,
    "encryptionScheme" : null,
    "escapeMarkup" : null,
    "externalDeveloperName" : null,
    "externalId" : false,
    "formula" : null,
    "formulaTreatBlanksAs" : null,
    "inlineHelpText" : null,
    "isAIPredictionField" : null,
    "isConvertLeadDisabled" : null,
    "isFilteringDisabled" : null,
    "isNameField" : null,
    "isSortingDisabled" : null,
    "label" : "Community (Region)",
    "length" : null,
    "lookupFilter" : null,
    "maskChar" : null,
    "maskType" : null,
    "metadataRelationshipControllingField" : null,
    "populateExistingRows" : null,
    "precision" : null,
    "readOnlyProxy" : null,
    "referenceTargetField" : null,
    "referenceTo" : null,
    "relationshipLabel" : null,
    "relationshipName" : null,
    "relationshipOrder" : null,
    "reparentableMasterDetail" : null,
    "required" : false,
    "restrictedAdminField" : null,
    "scale" : null,
    "securityClassification" : null,
    "startingNumber" : null,
    "stripMarkup" : null,
    "summarizedField" : null,
    "summaryFilterItems" : null,
    "summaryForeignKey" : null,
    "summaryOperation" : null,
    "trackFeedHistory" : false,
    "trackHistory" : false,
    "trackTrending" : null,
    "translateData" : null,
    "type" : "Picklist",
    "unique" : null,
    "urls" : null,
    "valueSet" : {
      "controllingField" : null,
      "restricted" : null,
      "valueSetDefinition" : {
        "sorted" : false,
        "value" : [ {
          "color" : null,
          "default" : false,
          "description" : null,
          "isActive" : null,
          "label" : "Canada",
          "urls" : null,
          "valueName" : "Canada"
        }, {
          "color" : null,
          "default" : false,
          "description" : null,
          "isActive" : null,
          "label" : "US",
          "urls" : null,
          "valueName" : "US"
        },
         {
          "color" : null,
          "default" : false,
          "description" : null,
          "isActive" : null,
          "label" : "UKk",
          "urls" : null,
          "valueName" : "UK"
        } 
       ]
      },
      "valueSetName" : null,
      "valueSettings" : null
    },
    "visibleLines" : null,
    "writeRequiresMasterRead" : null
  },
  "FullName" : "Case.Community_Group__c"
}

Here is JSON Body that has to be sent as request body in this callout: https://github.com/ayub-ansari/Create-Picklist-Field-Using-Apex/blob/master/FirstHTTPRequestToUpdatePicklistField