diff --git a/eslint/babel-eslint-parser/package.json b/eslint/babel-eslint-parser/package.json
index 2cd150eab58e..a4f1ed1a0c96 100644
--- a/eslint/babel-eslint-parser/package.json
+++ b/eslint/babel-eslint-parser/package.json
@@ -23,7 +23,7 @@
     "./package.json": "./package.json"
   },
   "peerDependencies": {
-    "@babel/core": ">=7.10.0",
+    "@babel/core": ">=7.11.0",
     "eslint": ">=7.5.0"
   },
   "dependencies": {
diff --git a/eslint/babel-eslint-parser/src/index.js b/eslint/babel-eslint-parser/src/index.js
index 2da046b0f05f..e4be37d822ff 100644
--- a/eslint/babel-eslint-parser/src/index.js
+++ b/eslint/babel-eslint-parser/src/index.js
@@ -12,19 +12,22 @@ import convert from "./convert";
 import analyzeScope from "./analyze-scope";
 import visitorKeys from "./visitor-keys";
 
-let isRunningSupportedVersion;
+let isRunningMinSupportedCoreVersion = null;
 
 function baseParse(code, options) {
-  if (typeof isRunningSupportedVersion !== "boolean") {
-    isRunningSupportedVersion = semver.satisfies(
+  // Ensure we're using a version of `@babel/core` that includes `parse()` and `tokTypes`.
+  const minSupportedCoreVersion = ">=7.2.0";
+
+  if (typeof isRunningMinSupportedCoreVersion !== "boolean") {
+    isRunningMinSupportedCoreVersion = semver.satisfies(
       babelCoreVersion,
-      packageJson.peerDependencies["@babel/core"],
+      minSupportedCoreVersion,
     );
   }
 
-  if (!isRunningSupportedVersion) {
+  if (!isRunningMinSupportedCoreVersion) {
     throw new Error(
-      `@babel/eslint-parser@${packageJson.version} does not support @babel/core@${babelCoreVersion}. Please upgrade to @babel/core@${packageJson.peerDependencies["@babel/core"]}`,
+      `@babel/eslint-parser@${packageJson.version} does not support @babel/core@${babelCoreVersion}. Please upgrade to @babel/core@${minSupportedCoreVersion}.`,
     );
   }