diff --git a/packages/babel-parser/src/parser/expression.js b/packages/babel-parser/src/parser/expression.js
index 74d09fd8a05f..9df262f9cf13 100644
--- a/packages/babel-parser/src/parser/expression.js
+++ b/packages/babel-parser/src/parser/expression.js
@@ -2306,20 +2306,23 @@ export default class ExpressionParser extends LValParser {
   parseIdentifierName(pos: number, liberal?: boolean): string {
     let name: string;
 
-    if (this.match(tt.name)) {
+    const { start, type } = this.state;
+
+    if (type === tt.name) {
       name = this.state.value;
-    } else if (this.state.type.keyword) {
-      name = this.state.type.keyword;
+    } else if (type.keyword) {
+      name = type.keyword;
 
       // `class` and `function` keywords push function-type token context into this.context.
       // But there is no chance to pop the context if the keyword is consumed
       // as an identifier such as a property name.
-      const context = this.state.context;
+      const curContext = this.curContext();
       if (
-        (name === "class" || name === "function") &&
-        context[context.length - 1].token === "function"
+        (type === tt._class || type === tt._function) &&
+        (curContext === ct.functionStatement ||
+          curContext === ct.functionExpression)
       ) {
-        context.pop();
+        this.state.context.pop();
       }
     } else {
       throw this.unexpected();
@@ -2330,12 +2333,7 @@ export default class ExpressionParser extends LValParser {
       // This will prevent this.next() from throwing about unexpected escapes.
       this.state.type = tt.name;
     } else {
-      this.checkReservedWord(
-        name,
-        this.state.start,
-        !!this.state.type.keyword,
-        false,
-      );
+      this.checkReservedWord(name, start, !!type.keyword, false);
     }
 
     this.next();
diff --git a/packages/babel-parser/src/tokenizer/context.js b/packages/babel-parser/src/tokenizer/context.js
index a53ec3d11040..f93c19971efb 100644
--- a/packages/babel-parser/src/tokenizer/context.js
+++ b/packages/babel-parser/src/tokenizer/context.js
@@ -104,10 +104,7 @@ tt.incDec.updateContext = function () {
 };
 
 tt._function.updateContext = tt._class.updateContext = function (prevType) {
-  if (prevType === tt.dot || prevType === tt.questionDot) {
-    // when function/class follows dot/questionDot, it is part of
-    // (optional)MemberExpression, then we don't need to push new token context
-  } else if (
+  if (
     prevType.beforeExpr &&
     prevType !== tt.semi &&
     prevType !== tt._else &&
@@ -134,10 +131,6 @@ tt.backQuote.updateContext = function () {
   this.state.exprAllowed = false;
 };
 
-tt.star.updateContext = function () {
-  this.state.exprAllowed = false;
-};
-
 // we don't need to update context for tt.braceBarL because we do not pop context for tt.braceBarR
 tt.braceHashL.updateContext = function () {
   this.state.context.push(types.recordExpression);
diff --git a/packages/babel-parser/src/tokenizer/types.js b/packages/babel-parser/src/tokenizer/types.js
index 535e16bd1204..ca67cb0732cf 100644
--- a/packages/babel-parser/src/tokenizer/types.js
+++ b/packages/babel-parser/src/tokenizer/types.js
@@ -155,7 +155,8 @@ export const types: { [name: string]: TokenType } = {
   plusMin: new TokenType("+/-", { beforeExpr, binop: 9, prefix, startsExpr }),
   // startsExpr: required by v8intrinsic plugin
   modulo: new TokenType("%", { beforeExpr, binop: 10, startsExpr }),
-  star: createBinop("*", 10),
+  // unset `beforeExpr` as it can be `function *`
+  star: new TokenType("*", { binop: 10 }),
   slash: createBinop("/", 10),
   exponent: new TokenType("**", {
     beforeExpr,