Are partial F test and student test the same for one parameter? In the context of a multiple linear regression, is applying a partial F-test on the significance of a single parameter equivalent to applying a student test on this parameter?

Yes, those are equivalent. You can find this shown in a good regression textbook, such as Agresti (2015).
Let’s see this in a simulation to convince ourselves that it is true, too.
set.seed(2022) 
N <- 30
x1 <- runif(N)
x2 <- runif(N)
x3 <- rbinom(N, 1, 0.5)
y <- x1 + x2 - x3 + rnorm(N)
L_full <- lm(y ~ x1 + x2 + x3)
L1     <- lm(y ~      x2 + x3)
L2     <- lm(y ~ x1      + x3)
L3     <- lm(y ~ x1 + x2     )
summary(L_full)

anova(L_full, L1)
anova(L_full, L2)
anova(L_full, L3)

In this simulation, I get the same p-values either way.
Agresti, Alan. Foundations of linear and generalized linear models. John Wiley & Sons, 2015.
