How to use Palladio font in text body but Computer Modern for Equations?

Don't load the mathpazo package. Instead, load the newpxtext package. As the name of the package indicates, it provides only text fonts, no math fonts.

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A full MWE:

\documentclass[12pt,a4paper]{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{lipsum}

%%\usepackage[sc]{mathpazo}
\usepackage{newpxtext}
\linespread{1.05}  

\begin{document}
\lipsum[2]
\begin{equation}
  f(x)=x^2 \int_{-\infty}^{\infty} \sin(2\pi f_0 t)\,dt
\end{equation}
\end{document}

Just rename the default family name:

\documentclass[12pt,a4paper]{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{lipsum}

\renewcommand{\rmdefault}{pplj}

\linespread{1.05}

\begin{document}

\textsc{Abcdef}

\lipsum[1]

\begin{equation}
  f(x)=x^2 \cdot \int_{-\infty}^{\infty} \sin(2\pi f_0 t)
\end{equation}
\end{document}

The output of pdffonts is

name                                 type              encoding         emb sub uni object ID
------------------------------------ ----------------- ---------------- --- --- --- ---------
DZRMSF+URWPalladioL-Roma             Type 1            Custom           yes yes no       4  0
CXJMMY+TeXPalladioL-SC               Type 1            Custom           yes yes no       5  0
NMBJYD+CMMI12                        Type 1            Builtin          yes yes no       6  0
VRSLZY+CMR12                         Type 1            Builtin          yes yes no       7  0
VOPKHO+CMR8                          Type 1            Builtin          yes yes no       8  0
CKUQXN+CMSY10                        Type 1            Builtin          yes yes no       9  0
UYVLXV+CMEX10                        Type 1            Builtin          yes yes no      10  0
BECFBN+CMSY8                         Type 1            Builtin          yes yes no      11  0

With pplx instead of pplj you get “uppercase digits” and true small caps (thanks to Ralf Steubner for suggesting it).

The output of pdffonts

name                                 type              encoding         emb sub uni object ID
------------------------------------ ----------------- ---------------- --- --- --- ---------
DKRBIK+URWPalladioL-Roma             Type 1            Custom           yes yes no       4  0
GDZRLO+TeXPalladioL-SC               Type 1            Custom           yes yes no       5  0
NMBJYD+CMMI12                        Type 1            Builtin          yes yes no       6  0
VRSLZY+CMR12                         Type 1            Builtin          yes yes no       7  0
VOPKHO+CMR8                          Type 1            Builtin          yes yes no       8  0
CKUQXN+CMSY10                        Type 1            Builtin          yes yes no       9  0
UYVLXV+CMEX10                        Type 1            Builtin          yes yes no      10  0
BECFBN+CMSY8                         Type 1            Builtin          yes yes no      11  0

The output is bad in either case, as Palatino/Palladio is visually incompatible with Computer Modern.