diff --git a/cpp/src/parquet/thrift_internal.h b/cpp/src/parquet/thrift_internal.h
index 888f286f17648..cd424914f1110 100644
--- a/cpp/src/parquet/thrift_internal.h
+++ b/cpp/src/parquet/thrift_internal.h
@@ -257,7 +257,7 @@ static inline EncodedGeoStatistics FromThrift(
 }
 
 static inline format::EdgeInterpolationAlgorithm::type ToThrift(
-    LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm) {
+    LogicalType::EdgeInterpolationAlgorithm algorithm) {
   switch (algorithm) {
     case LogicalType::EdgeInterpolationAlgorithm::SPHERICAL:
       return format::EdgeInterpolationAlgorithm::SPHERICAL;
@@ -270,11 +270,12 @@ static inline format::EdgeInterpolationAlgorithm::type ToThrift(
     case LogicalType::EdgeInterpolationAlgorithm::KARNEY:
       return format::EdgeInterpolationAlgorithm::KARNEY;
     default:
-      throw ParquetException("Unknown value for geometry algorithm: ", algorithm);
+      throw ParquetException("Unknown value for geometry algorithm: ",
+                             static_cast<int>(algorithm));
   }
 }
 
-static inline LogicalType::EdgeInterpolationAlgorithm::algorithm FromThrift(
+static inline LogicalType::EdgeInterpolationAlgorithm FromThrift(
     const format::EdgeInterpolationAlgorithm::type algorithm) {
   switch (algorithm) {
     case format::EdgeInterpolationAlgorithm::SPHERICAL:
@@ -288,7 +289,8 @@ static inline LogicalType::EdgeInterpolationAlgorithm::algorithm FromThrift(
     case format::EdgeInterpolationAlgorithm::KARNEY:
       return LogicalType::EdgeInterpolationAlgorithm::KARNEY;
     default:
-      throw ParquetException("Unknown value for geometry algorithm: ", algorithm);
+      throw ParquetException("Unknown value for geometry algorithm: ",
+                             static_cast<int>(algorithm));
   }
 }
 
diff --git a/cpp/src/parquet/types.cc b/cpp/src/parquet/types.cc
index 1ace995a964bf..ee583185502f6 100644
--- a/cpp/src/parquet/types.cc
+++ b/cpp/src/parquet/types.cc
@@ -493,7 +493,7 @@ std::shared_ptr<const LogicalType> LogicalType::FromThrift(
       crs = type.GEOGRAPHY.crs;
     }
 
-    LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm;
+    LogicalType::EdgeInterpolationAlgorithm algorithm;
     if (!type.GEOGRAPHY.__isset.algorithm) {
       algorithm = LogicalType::EdgeInterpolationAlgorithm::SPHERICAL;
     } else {
@@ -563,7 +563,7 @@ std::shared_ptr<const LogicalType> LogicalType::Geometry(std::string crs) {
 }
 
 std::shared_ptr<const LogicalType> LogicalType::Geography(
-    std::string crs, LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm) {
+    std::string crs, LogicalType::EdgeInterpolationAlgorithm algorithm) {
   return GeographyLogicalType::Make(std::move(crs), algorithm);
 }
 
@@ -1723,7 +1723,7 @@ bool LogicalType::Impl::Geometry::Equals(const LogicalType& other) const {
 }
 
 const std::string& GeometryLogicalType::crs() const {
-  return (dynamic_cast<const LogicalType::Impl::Geometry&>(*impl_)).crs();
+  return checked_cast<const LogicalType::Impl::Geometry&>(*impl_).crs();
 }
 
 std::shared_ptr<const LogicalType> GeometryLogicalType::Make(std::string crs) {
@@ -1743,9 +1743,7 @@ class LogicalType::Impl::Geography final : public LogicalType::Impl::Incompatibl
   bool Equals(const LogicalType& other) const override;
 
   const std::string& crs() const { return crs_; }
-  LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm() const {
-    return algorithm_;
-  }
+  LogicalType::EdgeInterpolationAlgorithm algorithm() const { return algorithm_; }
 
   std::string_view algorithm_name() const {
     switch (algorithm_) {
@@ -1765,14 +1763,14 @@ class LogicalType::Impl::Geography final : public LogicalType::Impl::Incompatibl
   }
 
  private:
-  Geography(std::string crs, LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm)
+  Geography(std::string crs, LogicalType::EdgeInterpolationAlgorithm algorithm)
       : LogicalType::Impl(LogicalType::Type::GEOGRAPHY, SortOrder::UNKNOWN),
         LogicalType::Impl::SimpleApplicable(parquet::Type::BYTE_ARRAY),
         crs_(std::move(crs)),
         algorithm_(algorithm) {}
 
   std::string crs_;
-  LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm_;
+  LogicalType::EdgeInterpolationAlgorithm algorithm_;
 };
 
 std::string LogicalType::Impl::Geography::ToString() const {
@@ -1832,8 +1830,7 @@ const std::string& GeographyLogicalType::crs() const {
   return (dynamic_cast<const LogicalType::Impl::Geography&>(*impl_)).crs();
 }
 
-LogicalType::EdgeInterpolationAlgorithm::algorithm GeographyLogicalType::algorithm()
-    const {
+LogicalType::EdgeInterpolationAlgorithm GeographyLogicalType::algorithm() const {
   return (dynamic_cast<const LogicalType::Impl::Geography&>(*impl_)).algorithm();
 }
 
@@ -1842,7 +1839,7 @@ std::string_view GeographyLogicalType::algorithm_name() const {
 }
 
 std::shared_ptr<const LogicalType> GeographyLogicalType::Make(
-    std::string crs, LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm) {
+    std::string crs, LogicalType::EdgeInterpolationAlgorithm algorithm) {
   auto* logical_type = new GeographyLogicalType();
   logical_type->impl_.reset(new LogicalType::Impl::Geography(std::move(crs), algorithm));
   return std::shared_ptr<const LogicalType>(logical_type);
diff --git a/cpp/src/parquet/types.h b/cpp/src/parquet/types.h
index fbb7666b5c82e..2d6c48ff25a4c 100644
--- a/cpp/src/parquet/types.h
+++ b/cpp/src/parquet/types.h
@@ -168,15 +168,13 @@ class PARQUET_EXPORT LogicalType {
     enum unit { UNKNOWN = 0, MILLIS = 1, MICROS, NANOS };
   };
 
-  struct EdgeInterpolationAlgorithm {
-    enum algorithm {
-      UNKNOWN = 0,
-      SPHERICAL = 1,
-      VINCENTY = 2,
-      THOMAS = 3,
-      ANDOYER = 4,
-      KARNEY = 5
-    };
+  enum class EdgeInterpolationAlgorithm {
+    UNKNOWN = 0,
+    SPHERICAL = 1,
+    VINCENTY = 2,
+    THOMAS = 3,
+    ANDOYER = 4,
+    KARNEY = 5
   };
 
   /// \brief If possible, return a logical type equivalent to the given legacy
@@ -229,7 +227,7 @@ class PARQUET_EXPORT LogicalType {
   static std::shared_ptr<const LogicalType> Geometry(std::string crs = "");
 
   static std::shared_ptr<const LogicalType> Geography(
-      std::string crs = "", LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm =
+      std::string crs = "", LogicalType::EdgeInterpolationAlgorithm algorithm =
                                 EdgeInterpolationAlgorithm::SPHERICAL);
 
   /// \brief Create a placeholder for when no logical type is specified
@@ -480,11 +478,11 @@ class PARQUET_EXPORT GeometryLogicalType : public LogicalType {
 class PARQUET_EXPORT GeographyLogicalType : public LogicalType {
  public:
   static std::shared_ptr<const LogicalType> Make(
-      std::string crs = "", LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm =
+      std::string crs = "", LogicalType::EdgeInterpolationAlgorithm algorithm =
                                 EdgeInterpolationAlgorithm::SPHERICAL);
 
   const std::string& crs() const;
-  LogicalType::EdgeInterpolationAlgorithm::algorithm algorithm() const;
+  LogicalType::EdgeInterpolationAlgorithm algorithm() const;
   std::string_view algorithm_name() const;
 
  private:
diff --git a/python/pyarrow/_parquet.pxd b/python/pyarrow/_parquet.pxd
index 24332e9ba01e0..9a65e0df16527 100644
--- a/python/pyarrow/_parquet.pxd
+++ b/python/pyarrow/_parquet.pxd
@@ -78,7 +78,7 @@ cdef extern from "parquet/api/schema.h" namespace "parquet" nogil:
         ParquetTimeUnit_MICROS" parquet::LogicalType::TimeUnit::MICROS"
         ParquetTimeUnit_NANOS" parquet::LogicalType::TimeUnit::NANOS"
 
-    enum ParquetEdgeInterpolationAlgorithm" parquet::LogicalType::EdgeInterpolationAlgorithm::algorithm":
+    enum ParquetEdgeInterpolationAlgorithm" parquet::LogicalType::EdgeInterpolationAlgorithm":
         ParquetEdgeInterpolationAlgorithm_UNKNOWN" parquet::LogicalType::EdgeInterpolationAlgorithm::UNKNOWN"
         ParquetEdgeInterpolationAlgorithm_SPHERICAL" parquet::LogicalType::EdgeInterpolationAlgorithm::SPHERICAL"
         ParquetEdgeInterpolationAlgorithm_VINCENTY" parquet::LogicalType::EdgeInterpolationAlgorithm::VINCENTY"