<p>In this paper, we study the boundary behavior of Milnor’s parameterization <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Phi : \mathcal {B}_d\rightarrow \mathcal {H}_d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <msub> <mi mathvariant="script">B</mi> <mi>d</mi> </msub> <mo stretchy="false">→</mo> <msub> <mi mathvariant="script">H</mi> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of the central hyperbolic component <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {H}_d\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> via Blaschke products. We establish a boundary extension theorem by giving a necessary and sufficient condition for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(D\in \partial \mathcal {B}_d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>∈</mo> <mi>∂</mi> <msub> <mi mathvariant="script">B</mi> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> which allows <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-extension. Further we show that cusps are dense in a full Hausdorff dimensional subset of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\partial \mathcal {H}_d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <msub> <mi mathvariant="script">H</mi> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, partially answering a question of McMullen.</p>