<p>This paper is a further study on parallel sum of operators. A sufficient condition for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1973_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </InlineMediaObject> <EquationSource Format="TEX">\((A:B)^{\frac{1}{2}}\ge A^{\frac{1}{2}}:B^{\frac{1}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo>≥</mo> <msup> <mi>A</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo>:</mo> <msup> <mi>B</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> </mrow> </math></EquationSource> </InlineEquation> through the solution of the Lyapunov equation is provided, where <i>A</i> and <i>B</i> are positive operators, <i>A</i>&#xa0;:&#xa0;<i>B</i> denotes the parallel sum of <i>A</i> and <i>B</i>. In addition, the weakly parallel sum of operators is studied, in which the ranges of operators are not necessarily to be closed. A sufficient condition for the distribution law of weakly parallel sum is given, and the existence of solution to the weakly parallel sum equation <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1973_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(A:X=C\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>:</mo> <mi>X</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation> is discussed.</p>