Clocks on the surface of each planet tick at different rates. It is a small effect, amounting to a moderate number of parts per billion.
I used Mathematica to calculate the time dilation on the surface of the Sun and the surface of the planets, relative to a clock far from the solar system that is stationary relative to the Sun. (Mathematica has data on lots of physical quantities.)
I took into account both gravitational time dilation and kinematic time dilation, using the formula for the time dilation factor,
$$\sqrt{1-\frac{2\phi}{c^2}-\left(1-\frac{2\phi}{c^2}\right)^{-1}\frac{v^2}{c^2}}\approx 1-\frac{\phi}{c^2}-\frac{1}{2}\frac{v^2}{c^2}.$$
Here $\phi$ is the positive-ized Newtonian gravitional potential,
$$\phi=\sum_i\frac{G M_i}{R_i},$$
and $v$ is the orbital speed.
For the Sun, I included only the gravitational potential from itself. For each planet, I included the gravitational potential from the planet and from the Sun, but not from other planets.
For calculating the orbital speed, I approximated the planetary orbits as circular, with an orbital radius equal to the average of the semimajor and semiminor axes.
Of course, all the results are very close to 1, but a bit smaller. The following table expresses the results as how much less than 1 the time dilation factor is, in parts per billion.
$$\begin{array}{ccccc}
\text{Sun} & 2122. & 2122. & 0 & 0 \\
\text{Mercury} & 38.35 & 0.1005 & 25.50 & 12.75 \\
\text{Venus} & 21.07 & 0.5972 & 13.65 & 6.823 \\
\text{Earth} & 15.50 & 0.6961 & 9.870 & 4.935 \\
\text{Mars} & 9.86 & 0.1406 & 6.478 & 3.239 \\
\text{Jupiter} & 23.01 & 20.16 & 1.897 & 0.9485 \\
\text{Saturn} & 8.80 & 7.247 & 1.0350 & 0.5175 \\
\text{Uranus} & 3.313 & 2.542 & 0.5143 & 0.2572 \\
\text{Neptune} & 3.58 & 3.089 & 0.3283 & 0.1641 \\
\end{array}$$
The first numeric column is the body's total time dilation on its surface. The other three show the breakdown into gravitational dilation due to the body's own gravity; gravitational dilation due to the gravity of other bodies (for the planets, the Sun); and kinematic dilation due to orbital motion around the Sun.