To remove the facet labels completely, you can use something similar to the following R code:

library(ggpubr)
# Facet default 
p <- ggplot(ToothGrowth, aes(x = supp, y = len)) +
  geom_boxplot() +
  theme_bw() +
  facet_wrap(~dose)
p

# Remove facet title
p + theme(
  strip.background = element_blank(),
  strip.text.x = element_blank()
)

In the current version, ggpubr doesn’t support the argument labeller = “label_parsed”, so you can go as follow:

# Plot
library(ggpubr)
ggbarplot(
  df, x = "supp", y = "len", add = "mean_sd",
  facet.by = "dose"
  ) +
   facet_wrap(~dose, labeller = label_parsed)

You want labeller = label_parsed. Here’s a simple example

# Data preparation
data("ToothGrowth")
df <- ToothGrowth

df$dose <- factor(
  df$dose,
  levels = c("0.5", "1", "2"),
  labels = c(
    "0.5~(mg~L^{-1})", 
    "1~(mg~L^{-1})", 
    "2~(mg~L^{-1})"
  ))

# Plot
library(ggplot2)
ggplot(df, aes(x = supp, y = len)) + 
  geom_boxplot()+
  facet_wrap(~dose, labeller = label_parsed)