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In this note, we investigate the pinching problem for oriented compact submanifolds of dimension <i>n</i> with parallel normalized mean curvature vector in a space form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>≥</mo><mi>c</mi></mrow></semantics></math></inline-formula>, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even <i>n</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>.